chapter 12: quadratic functions

T H E M E : Gravity

Even very small children understand an important law of physics: When youdrop something, it falls. But what makes the object fall? Scientists have

named the force gravity. Gravity is measured by hanging the object on a springscale. This measure is called the weight of the object.

By working to understand and measure gravity, scientists have succeeded inovercoming its effects.

• Pilots (page 529) command airplanes, jets, helicopters, and spacecraft.Pilots must understand how speed, altitude, temperature, and the weightof the plane, including its contents, affect air travel.

• Air Traffic Controllers (page 549) ensure safe air travel by monitoringthe movement of aircraft. Controllers use radar and visual observation tomonitor the progress of aircraft. They work together to make sure planesstay a safe distance apart.

516

QuadraticFunctions

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Chapter 12 Quadratic Functions

Use the table for Questions 1–4.

1. A tool weighs 2.5 lb on Earth. What is the weight of the tool onMars? on Jupiter?

2. An astronaut has two oxygen tanks. On Earth, Tank A weighstwice as much as Tank B. If the two tanks are transported toSaturn, Tank A’s weight will be how many times the weight ofTank B?

3. At which of the locations shown in the table would you weigh lessthan you do on Earth?

4. A bag of moon rocks weighs 225 lb on the moon. To the nearesttenth, what is the weight of the bag of rocks on Earth?

CHAPTER INVESTIGATIONA child’s wagon has no engine or other visible means of moving itself forward. Yet, when the wagon is positioned at the top of a steep hill and begins to move down the hill, its speed increases as it goes. The wagon is propelled by gravity.

Working TogetherDesign a small gravity-driven vehicle weighing no more than 10 oz.Time the vehicle’s descent down an incline, recording the angle ofdescent and the time. Explore how changing the shape or weight ofthe vehicle affects its speed. Use the Chapter Investigation icons toguide your group.

Data Activity: How Does Gravity Affect Weight?

517

How Does GravityAffect Weight?Location

Mercury

Venus

Earth

Moon

Mars

Jupiter

Saturn

Uranus

Neptune

Pluto

2842 lb

9069 lb

10,000 lb

1656 lb

3803 lb

23,394 lb

9253 lb

7944 lb

11,247 lb

408 lb

Weight of 5-ton elephanton Earth

0.95075 lb; 5.8485 lb

2 times

Mercury, Venus, Moon, Mars, Saturn,Uranus, Pluto

1358.7 lb

The skills on these two pages are ones you have already learned. Review theexamples and complete the exercises. For additional practice on these and moreprerequisite skills, see pages 654–661.

ADDITION AND SUBTRACTION

Making a table of ordered pairs to help graph a linear equation is a skill that willbe especially useful as you learn to graph quadratic functions.

Example Graph the equation 2x � y � 8.

Make a table of values. Generally, use �3 to 3 for the value of x.

Find the value of y foreach value of x by

substituting the valueof x in the equation.

2(�3) � y � 82(�2) � y � 8

Solve each equation fory and fill in the table.

With these ordered pairs you can graph the line of the equation.

Make a table of values for each equation. Round to the nearest thousandth if necessary.

1. y � �12

�x � 3 2. 3y � 2x � 4 3. 4x � 2y � 1

4. �2y � x � 3 5. y � 3x � 7 6. y � x � �6

7. 2x � 3y � �6 8. 4 � 3x � �12

�y 9. 5x � y � �8

10. 4y � x � �7 11. 8 � 3y � 2x 12. 2x � 2y � �2

GRAPHS OF FUNCTIONS

You can easily tell whether a graph is that of a function or not by the Vertical LineTest. When a vertical line is drawn through the graph of a relation, the relation isnot a function if the vertical line intersects the graph in more than one point.

Graph of a function: Not the graph of a function:

�5

�5 5

x

y

�5

5

5

x

y


12CH

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518

12 Are You Ready?Refresh Your Math Skills for Chapter 12

For 1–12, see additional answers.

x y�3 14�2 12�1 10

0 81 62 43 2

x y�3�2�1

0123

519Chapter 12 Are You Ready?

Use the Vertical Line Test to determine if each relation is a function.13. 14.

15. 16.

PYTHAGOREAN THEOREM

You have used the Pythagorean Theorem to find measures of the sides of right triangles. It is a very valuable formulato know, and one you will use in real life.

a 2 � b 2 � c 2

Find the missing side measures to the nearest hundredth.

17. 18. 19. 20.

21. 22. 23. 24. 7

5

55

8

10

4

6

3817

9

15

7

8

9

13

a

b

c

�5

5

5�5 x

y

�5

�5 5

5

x

y

�5

�5 5

5

x

y

�5

�5 5

5

x

y

yes no

nono

7.21

12.81

8.607.07

15.8110.63

12

33.99

You will need a graphing calculator for this activity.

GRAPHING Use the ZOOM menu to make sure your display window is set onstandard size. Then graph each of the following equations.

a. y � �x b. y � x 2 � 2x � 1 c. y � x � 3

d. y � �x 2 � 1 e. y � 2x � 2 f. y � 2x 2

1. How can you tell by looking at an equation whether the graph will be a line ora curve?

2. Using the equations above, write two equations: one for a straight line andone for a curve.

BUILD UNDERSTANDING

In this section you will learn to graph equations that containsecond degree or quadratic terms.

A quadratic equation in x contains an x 2 term and involves no termwith a higher power of x. The simplest quadratic equation is y � x 2.

E x a m p l e 1

Graph y � x 2.

SolutionFind at least five ordered pairs by selecting x-valuesand solving the equation to find y-values.

Graph the ordered pairs.

Draw a smooth curve through the points.

Because there is only one y-value for each x-value, this is the graph of a function.When the domain of a quadratic function is the set of real numbers, the graph isa parabola.

Notice that there is more than one x-value for each y-value. There are two x-values for each y-value except for point (0, 0), the lowest point on the parabola.

Chapter 12 Quadratic Functions520

12-1 Graph ParabolasGoals ■ Graph parabolas or second degree equations.

Applications Geology, Small Business, Physics

x �3 �1 0 1 3y 9 1 0 1 9

8

6

4

2

2 4�2�4

y

x

CheckUnderstanding

Which of the followingare quadratic equations?

1. y � x3 � 6x2

2. y � 55 � x2

3. y � 4x � 16

4. y � 5x2 � 9x � 1

Equations that contain an x2 term will have curved graphs.

2 and 4

The parabola in Example 1 opens upward. For some functions, the parabolasopen downward.

E x a m p l e 2

Graph y � �2x 2.

SolutionMake a table of ordered pairs.

Graph the points corresponding to the orderedpairs and draw a smooth curve through them.

The vertex is the lowest point on a parabola that opens upward, and the highestpoint on a parabola that opens downward. The graphs for y � x 2 and y � �2x 2

both have the point (0, 0) as their vertex.

E x a m p l e 3

GEOLOGY The distribution of a trace element within a geologicsample can be modeled by the equation y � 3x 2 � 2. Graph andlocate the vertex of the parabola.

SolutionMake a table of ordered pairs.

Graph the points and draw a smooth curve. Look for a y-value that has only one x-value. The vertex is (0, �2).

GRAPHING You can use a graphing calculator to find the coordinates of thevertex. Key in the quadratic equation and graph. You may have to use the zoomfeatures to adjust the size of the display. Press the TRACE key. Your calculator willplace a point at the y-intercept. If the y-axis is the line of symmetry, thecoordinates for the y-intercept are also the coordinates for the vertex. Use agraphing calculator to locate the vertex of the parabola in Example 3.

You can use the arrow keys to move the trace point to locate the vertex or otherpoints on the parabola. However, you must remember that there are limits to thedisplay capabilities of a graphing calculator. The zoom feature allows you to seemore detail.

The vertex lies on the line that divides the parabola in half. This line is called theaxis of symmetry of the parabola.

In each of the three previous examples, the parabola is divided in half by the y-axis, which is the line x � 0. The axis of symmetry is not always x � 0. It isdetermined by the given equation.

Lesson 12-1 Graph Parabolas 521

x �2 �1 0 1 2y �8 �2 0 �2 �8

�2

�4

�6

�8

2 4�2�4

y

x

x �2 �1 0 1 2y 10 1 �2 1 10

8

6

4

2

2 4�2�4

y

x

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TRY THESE EXERCISES

Copy and complete each table. Then draw the graph.

1. y � x 2 � 3 2. y � �x 2

Graph each function for the domain of real numbers. For each graph, give thecoordinates of the vertex.

3. y � x 2 � 2 4. y � �x 2 � 3 5. y � �5x 2

6. SMALL BUSINESS A study shows that the daily revenue from product salescan be modeled by the equation y � �5x 2 � 12, where y equals the revenuein hundreds of dollars and x equals possible increases and decreases in price.Graph the equation. What is the maximum revenue? (Hint: The y-coordinateof the vertex is the maximum revenue in hundreds of dollars.)

PRACTICE EXERCISES • For Extra Practice, see page 700.

Graph each function for the domain of real numbers. For each graph, give thecoordinates of the vertex.

7. y � 4x 2 8. y � 2x 2 � 2 9. y � �x 2 � 1

Determine if the graph of each equation below opens upward or downward.

10. y � �3x 2 � 4 11. y � 7x 2 12. y � �x 2 � 10

13. GRAPHING The equations below have the form y � ax 2. Graph eachequation on a graphing calculator. How does the graph change as the valueof a changes?

a. y � 10x 2 b. y � 4x 2 c. y � 0.5x 2

d. y � �0.5x 2 e. y � �3x 2 f. y � �15x 2

14. WRITING MATH What do you notice about the location of the vertex of aparabola that is the graph of an equation in the form y � ax 2? What do younotice about the location of the axis of symmetry of the graph?

15. GRAPHING The equations below are in the form y � ax 2 � c. Graph the equations on a graphing calculator. How does the graph change as the value of c decreases?

a. y � 2x 2 � 4 b. y � 2x 2 � 3 c. y � 2x 2 � 1

d. y � 2x 2 � 1 e. y � 2x 2 � 3 f. y � 2x 2 � 5

16. WRITING MATH What do you notice about the location of the vertex andaxis of symmetry of the parabola you obtain when you graph an equation in the form y � ax 2 � c?

17. PHYSICS When an object is dropped and falls to the ground under the force of gravity, its height, y, in feet, x seconds after being dropped is given by y � �16x2 � 18. Find the height from which it was dropped.

x �4 �2 0 2 4y


See additional answers for graphs.

13 1 �3 1 13 �36 �9 0 �9 �36


(0, 2) (0, �3) (0, 0)

$1200


(0, 0) (0, 2) (0, 1)

downwardupwarddownward

As the absolute value of a decreases the graph becomes wider.

The vertex is always at the origin; The axis of symmetry is the y-axis.

The graph shifts downward.

The vertex is always on the y-axis; the y-axis is the axis of symmetry.

18 ft

523

18. PHYSICS A projectile is shot vertically up in the air from ground level. Itsdistance d, in feet, after t seconds is given by d � 96t � 16t2. Find the valuesof t when d is 96 ft.

EXTENDED PRACTICE EXERCISES

Each graph below is for an equation of the form y � ax 2 � c. The value of ais �6 or �6 for one equation, and �1 or �1 for the other. Write the equation for each graph.

19. 20.

GRAPHING Use a graphing calculator. Graph each pair of equations in the samewindow. Find the vertex and axis of symmetry for each pair.

21. y � �x� 22. y � ��x�

y � ��x� y � � ��x�

The graph of y � a x 2 has a maximum point or a minimum point.

23. What form of equation has a graph with a maximum point?

24. What form of equation has a graph with a minimum point?

25. CHAPTER INVESTIGATION Gravity can be used to power a vehicle on aramp or incline. Work with your group to draw plans for a small vehicleweighing no more than 10 oz. As a first step, explore what types of materialsto use to build the vehicle. Gather materials and weigh samples of eachmaterial in its raw form. Once you have made a final selection of materials,draw a design for a simple gravity-powered vehicle. Your vehicle should havewheels or employ some other technology to reduce friction.

MIXED REVIEW EXERCISES

Simplify. (Lesson 11-1)

26. (6a � 3) � (4a � 6) 27. (2y 2 � y) � (5y � 8) 28. (4x 2 � 3x � 2) � (x � 4)

29. (3b 2 � 2b) � (4b � 3) 30. (6c � 2) � (5c � 8) 31. (�3d 2 � 8) � (4d � 9)

In each triangle, A�B� ‖ C�D�. Find x to the nearest tenth. (Lesson 7-5)

32. 33. 34.x

12

8

4

A B

C D

x

5

9

4

A B

C D

3

3

5

5

8

A B

C D

x

4

2

�2

�4

y

x�4 4

16

12

8

4

y

x1�1�2 2

Lesson 12-1 Graph Parabolas

about 1.3 sec and 4.7 sec

y � 6x2 � 2 y � �x2 � 3

(0, 0)

x-axis

An equation in which a � 0; the graph opens downward and has a maximum point.

An equation in which a � 0; the graph opens upward and has a minimum point.

10a � 3

3b2 � 2b � 3

2y2 � 4y � 8

11c � 6 �3d2 � 4d � 17

4x2 � 2x � 2

4 11.3 6

(0, 0)

x-axis

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Work with a partner to answer the following questions.

1. GRAPHING The following quadratic equations have the form y � ax 2 � bx. Graph the equations on a graphing calculator.

y � x 2 � 7x y � x 2 � 5x y � x 2 � 2x

y � x 2 � 7x y � x 2 � 5x y � x 2 � 2x

y � �x 2 � 7x y � �x 2 � 5x y � �x 2 � 2x

y � �x 2 � 7x y � �x 2 � 5x y � �x 2 � 2x

2. Copy and complete each sentence. Write � or � in each blank.

a. If a is ___?__ 0 and b is ___?__ 0, the vertex is in quadrant III.

b. If a is ___?__ 0 and b is ___?__ 0, the vertex is in quadrant IV.

c. If a is ___?__ 0 and b is ___?__ 0, the vertex is in quadrant I.

d. If a is ___?__ 0 and b is ___?__ 0, the vertex is in quadrant II.

BUILD UNDERSTANDING

In Lesson 12-1, you investigated quadratic equations of the formsy � ax 2 and y � ax 2 � c. You learned that for quadratic equations,the value of y is determined by the value of x; y is a function of x.This relationship is expressed as y � f(x).

As you have discovered, the graph of aquadratic equation is a parabola. Forthe parabolas you graphed in Lesson12-1, the vertex was always located onthe y-axis, and the y-axis was the axisof symmetry. Your exploration ofequations in the form y � ax 2 � bxshowed that other locations arepossible.

In this lesson, you will learn how to locate the vertex and axis ofsymmetry for any quadratic equation.

The general quadratic function may be written f (x) � ax 2 � bx � c ory � ax 2 � bx � c, where a, b, and c are real numbers and a � 0.


12-2 The General Quadratic FunctionGoal ■ Graph functions defined by the general quadratic

equation.

Applications Physics, Business, Astronomy

y

x

Reading Math

y � f(x) is read

“y equals f of x.”

CheckUnderstanding

In the general quadraticfunction, why can a notbe equal to 0?

� �

� �

� �

� �

If a � 0, the function islinear, not quadratic.

E x a m p l e 1

Graph y � 3x 2 � 4x � 1 on a graphing calculator. Estimate thecoordinates of the vertex.

SolutionEnter the equation. Graphthe function. Use the traceand zoom features tolocate the coordinates ofthe vertex.

The closer you zoom in,the closer the coordinateswill be to the actual valuesof x and y. Eventually, youmay be able to see arelationship between thedecimal values on the screen and a common fraction or wholenumber. For example, you may have arrived at the coordinates x � 0.6637119 and y � �2.333307.

0.6637119 is about �23

�.

2.333307 is about 2�13

�.

The vertex is approximately ��23

�, �2�13

��.

For a parabola defined by the equation y � ax 2 � bx � c, the vertex is always atthe point on the graph where the x-coordinate is x � ��

2ba�. The corresponding

y-value can be found by substituting the x-value into the equation.

E x a m p l e 2

Find the coordinates of the vertex for the graph of y � 3x 2 � 4x � 1.

Solutionx � ��

2ba�

x � ��2�

(34)

�� Substitute for a and b.

x � �46

� or �23

� Simplify.

y � 3��23

��2

� 4��23

�� 1 Substitute the x-value into the equation.

y � 3��49

�� 83

� � 1 Simplify.

y � �2�13

�

The coordinates of the vertex for the graph of y � 3x 2 � 4x � 1 are ��23

�, �2�13

��.

As you learned in Lesson 12-1, the axis of symmetry is a vertical line through thevertex of a parabola. The axis of symmetry for the graph of a quadratic function isx � ��

2ba�. For the graph of the function above, the axis of symmetry is x � �

23

�.

Lesson 12-2 The General Quadratic Function 525

5

3

1

�1

�3

1 2�1

y

x

X � .66371191 Y � �2.333307

CheckUnderstanding

Give the values of a, b,and c for each function.

1. y � 6x2 � 4x � 5

2. y � 7x2 � 3x � 2

3. y � 9x2 �6x � 5

4. y � 7 � x2

Technology Note

The zoom feature on agraphing calculator allowsyou to magnify a sectionof a graph. The Zoom Boxdefines the box to beenlarged.

To zoom in on the vertex:

1. Select Box from thezoom menu

2. Place the cursor on acorner of the area youwant to magnify; pressenter.

3. Move the cursor to thediagonally oppositecorner of the box; press enter.

1. 6, 4, 52. 7, �3, 23. 9, �6, �54. 1, 0, �7


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You can use the equation x � ��2ba� to graph a quadratic function.

E x a m p l e 3

Graph f (x) � �2x 2 � 3x � 1.

SolutionLocate the vertex.

x � ��2ba� � � ��2(

�

�

32)

�� 34

�

y � �2��34

��2

� 3��34

�� 1

� �2��196��

94

�� 1 � 2�18

�

The vertex is ��34

�, 2�18

��. The axis of symmetry is x � � �34

�.

Because a is less than 0, the parabola opens downward. Substitute values in theequation to locate a few points.

Use the axis of symmetry to visually locate other points. Draw a smooth curve.

As you know, a function in x assigns only one y-value for each x-value. To tellwhether a graph is of a function, check to see whether each vertical line in thecoordinate plane contains at most one point of the graph.

TRY THESE EXERCISES

GRAPHING Estimate the coordinates of the vertex for each parabola bygraphing the equation on a graphing calculator. Then use x � ��

2ba� to find

the coordinates.

1. y � x 2 � 4x � 3 2. y � 2x 2 � 3x � 1

3. 5x 2 � 2x � 5 � y 4. y � 15 � x 2 � 2x

Find the vertex and axis of symmetry. Then graph each equation.

5. y � �3x 2 � 9x � 1 6. y � 2x 2 � 8x � 3

7. y � 3x 2 � 2x 8. �x 2 � y � 3 � x


GRAPHING Estimate the coordinates of the vertex for each parabola bygraphing the equation on a graphing calculator. Then use x � ��

2ba� to find the

coordinates.

9. y � x 2 � 6x � 4 10. y � 3 � 2x 2 � 12x 11. �9 � y � �3x � 3x 2

12. y � x 2 � 4 13. �8x � 5 � �2x 2 � y 14. 4x 2 � 16x � 1 � y

x �1 0 1y 2 1 �4


�1

�3

�5

�7

y

x1�1�2

(2, �1)

��15

�, 4�45

�� (1, �16)

(�2, �5); x � �2

��12

�, 2�34

��; x � ��12

�

��32

�, 7�34

��; x � ��32

�

��13

�, ��13

��; x � ��13

�

(�3, �5)

(0, 4) (2, �3)(2, �15)

��12

�, 8�14

��(3, �15)

��34

�, �2�18

��

For 5–8, see additionalanswers for graphs.

Find the vertex and axis of symmetry.

15. y � 5x 2 � 10x � 1 16. x 2 � x � 3 � y 17. y � �x 2 � 1

18. BUSINESS The sales projections for a company can be represented by aquadratic equation in the form of y � ax 2 � bx � c for which c � 3, and theaxis of symmetry is x � �

56

�. Find the equation.

19. ASTRONOMY Observed movement in an object can be represented by aquadratic equation in the form y � ax2 � bx � c for which c � �2 and

the vertex is ��58

�, ��176��. Find the equation.

20. WRITING MATH Study graphs A and B shown at the right. Determine if each graph is the graph of a function. Explain your reasoning.


PHYSICS A cannon is fired at several differentangles. The paths of the cannonballs are shown on the graph.

21. For what firing angle did the cannonball reach the highest point?

22. For what angles did the cannonball travel about the same distance?

23. WRITING MATH What conclusion can you draw about the maximum rangeof the cannon?



24. (7bc)(3ab) 25. 6s 2(s 2 � 3s � 2) 26. (5sz 4)(3s 4z 5)

27. (2a 2)(4ac � 3bd) 28. (4r 2s 3)(2r 3s 2 � 8rs 4) 29. (25x 4y 2)(4x 3y 2 � 2x 2y 5)

Graph each function. (Lesson 2-3)

30. y � �23

�x � 8 31. y � 3x � 4 32. y � �2x � 3

33. y � �12

�x � 2 34. y � �3x � 1 35. y � �34

�x � 2

Lesson 12-2 The General Quadratic Function 527

70�55�

45�

35�

20�

4

2

�4

y

x2�2�4 4

4

2

�2

�4

y

x2�4 4

a. b.

(1, �4); x � 1 ��12

�, 2�34

��; x � ��12

�(0, 1); y-axis

y � 3x2 � 5x � 3 or y � �3x2 � 5x � 3

y � �4x2 � 5x � 2

See additional answers.

70°

20° and 70°, 35° and 55°

The maximum range is reached at about 45°.

21ab2c6s4 � 18s3 � 12s2

8r5s5 � 32r3s78a3c � 6a2bd

15s5z9

100x7y4 � 50x6y7

For 30–35, see additional answers for graphs.



PRACTICE LESSON 12-11. The relation described by y � x 2 is a function. Explain why.

2. Describe the vertex of:

a. a parabola that opens downward.

b. a parabola that opens upward.

Copy and complete each table. Then draw the graph.

3. y � x 2 � 4 4. y � 2x 2 � 1

Determine if the graph of each equation opens upward or downward.

5. y � �3x 2 � 4 6. y � 3x 2 � 8 7. y � 2x 2 � 4 8. y � �(x 2 � 2)

Use the graph of each function, where the domain is the real numbers, to namethe coordinates of the vertex.

9. y � x 2 � 6 10. y � 3x 2 � 1

11. y � �2x 2 � 1 12. y � �x 2 � 9

Tell whether the vertex of the graph is at the origin.

13. y � x 2 � 4 14. y � x 2 15. y � �x 2 16. y � �x 2 � 3

Tell whether the axis of symmetry of the graph is the y-axis.

17. y � � �2x� 18. y � 5x 2 � 3

19. y � � �5x � 3� 20. y � �3x 2 � 1

PRACTICE LESSON 12-221. What is the general form of a quadratic equation?

22. If f(x) � ax2 � bx � c is a quadratic function, can a � 0?

Estimate the coordinates of the vertex for each parabola by graphing the equation on a graphing calculator. Then use x � ��

2ba� to find the coordinates.

23. y � 2x 2 � 3x � 4 24. y � �3x 2 � 4x � 5

25. y � x 2 � 2x � 3 26. y � 4x 2 � 2x � 6

Find the vertex and axis of symmetry.

27. y � �2x 2 � x � 4 28. y � 3x 2 � 6x � 8 29. y � 2x 2 � 3x � 5

x �3 �2 �1 0 1 2 3y

x �3 �2 �1 0 1 2 3y 13


There is only one y-value foreach x-value.

the highest point on the parabola

the lowest point on the parabola


downward upward upward downward

(0, 1)

(0, 9)

(0, �6)

(0, 1)

no yes yes no

yes

yes

no

no

y � ax2 � bx � c

no

��34

�, �283��

(1, 2)

��23

�, �3�23

��

14

�, �5�34

��

��34

�, 3�78

��; x � �34

�(�1, �11); x � �1

27. ��14

�, �3�78

��; x � �14

�


Review and Practice Your Skills

Workplace Knowhow

PRACTICE LESSON 12-1–LESSON 12-2Use the graph of each function to name the coordinates of the vertex.(Lesson 12-1)

30. y � x 2 � 4x � 2 31. y � �2x 2 � x � 3 32. y � 3x 2 � 2x � 4

Graph each equation on a graphing calculator. Give the quadrant of the vertex.(Lesson 12-2)

33. y � �3x 2 � 3x � 7 34. y � 2x 2 � 5x � 8 35. y � �x 2 � x � 5

Find the vertex and axis of symmetry. (Lesson 12-2)

36. y � x 2 � 4x � 3 37. y � �3x 2 � 2x � 4 38. y � 2x 2 � 3x � 5

Chapter 12 Review and Practice Your Skills 529

Pilots are the commanders of airplanes, jets, helicopters and space shuttles.Pilots transport passengers and goods, fight fires, test new aircraft, monitor trafficand crime, dust crops and conduct rescue missions. Pilots must make manycalculations to fly safely. They must be able to read electronic instrumentsaccurately. For example, they use math to calculate the speed necessary for takeoff. To do so, they must consider many variables such as altitude of the airport,outside temperature, weight of the plane and speed and direction of wind.

One plane left New York headed to Tokyo flying at an average speed of 375 mi/h.Another plane left New York 1 h later following the same route and flying at anaverage speed of 500 mi/h. If both planes followed the same course, how manyhours after it left New York would the second plane catch up to the first plane?

1. Write an equation you could solve to answer the question.

2. Use a graphing utility to graph both sides of the equation and solve theproblem.

3. How many miles will each plane have flown when they are equidistant fromNew York?

4. A plane gained altitude at a rate of 1000 ft/min, descended 1300 ft, then started to climb again for 2�

14

� min at a rate of 800 ft/min. The total gain in altitude was 6000 ft. How long did the plane climb at a rate of 1000 ft/min?

5. An airplane takes off and climbs at a steady 18° angle. After flying along a pathof 2 mi, how much altitude has the plane gained in feet? Round to the nearestfoot.

(2, �2) ��14

�, �2�78

��

IV

��13

�, 4�13

��; x � ��13

� ��34

�, �6�18

��; x � �34

�

375 (x � 1) � 500x

Check student’s graph.

1500

5�12

� min

3263 ft

II I

��13

�, 3�23

��



(�2, �7); x � �2

Career – Pilots

mathmatters3.com/mathworks

http://mathmatters3.com/mathworks

GRAPHING Graph each equation below on a graphing calculator. How many x-intercepts, the points where the graph crosses the x-axis, does each graphhave? Use the trace feature, if necessary, to locate the x-intercepts.

a. y � x 2 � 25 b. y � 2x 2 � 3x � 1

c. y � 2x 2 � 3x � 5 d. y � x 2 � x

e. y � x 2 � 5 f. y � x 2 � 6x � 9

BUILD UNDERSTANDING

The x-intercepts are the solutions of the quadratic equation. They are the x-coordinates of points for which y � 0. As illustrated in the above activity, theremay be 0, 1, or 2 x-intercepts.

You can determine the number of x-intercepts and estimate their values bygraphing the quadratic equation. You can often find exact solutions by letting y � 0 and factoring the quadratic expression.

E x a m p l e 1

Solve 3x 2 � 6x � 0.

SolutionBy graphing:

Graph the related quadratic function y � 3x2 � 6x.

Locate the points where y � 0. Estimate the x-valuesfor these points. Use the zoom feature to more closelyestimate values. The x-values are the solutions.

The graph of the equation y � 3x 2 � 6x intersects thex-axis at two points. They are located approximatelyat x � 2 and x � 0.

By factoring:

To solve by factoring, let y � 0.

3x 2 � 6x � 03x(x � 2) � 0 Factor.3x � 0, x � 2 � 0 Solve each equation.x � 0, x � 2

The solutions for y � 3x 2 � 6x � 0 are 0 and 2.


12-3 Factor and GraphGoals ■ Use factoring to solve quadratic equations.

Applications Physics, Archery, Sports

9

7

5

1

�1

�3

1 3�1�3

y

x

2

0

0 1

2

1

If the parabola is tangent to the x-axis, there is only one solution to the quadratic equation.

E x a m p l e 2

Solve x 2 � 8x � 16 � 0.

SolutionWrite the related quadratic function.

y � x 2 � 8x � 16

By graphing:

Graph the equation y � x 2 � 8x � 16 todetermine the number of solutions. Locate thepoint where y � 0. The x-value is the solution.

The graph of the equation meets the x-axis atone point, approximately x � �4.

By factoring:

To solve by factoring, let y � 0.

x 2 � 8x � 16 � 0(x � 4)(x � 4) � 0

x � �4

The solution for x 2 � 8x � 16 � 0 is �4.

If the parabola for the equation does not meet the x-axis, the equation has nosolutions.

E x a m p l e 3

Solve � x 2 � x � 1 � 0.

SolutionGraph the related functionon a graphing calculator.

The graph of the equationdoes not cross the x-axis.

There are no solutions forthe equation y � �x 2 � x � 1.

Suppose a projectile is lauched from ground level. If you know the velocity withwhich the projectile is launched, you can find the time between launch andlanding using the equation vt � 16t 2 � 0, where v � velocity in feet per second(ft/sec) and t � time in seconds.

Lesson 12-3 Factor and Graph 531

40

30

20

�52�2�4�6�8�10

y

x

�10

�20

�30

�40

2�4�6 4 6

y

x

Mental Math Tip

An expression in the formax2 � bx � c in which a,b, and c are integers maybe factored if ac hasfactors with a sum of b.

Technology Note

If you have access tocharting and data analysissoftware, you may wantto investigate methods forgraphing functions. Formany programs, you enterdata and equations on aspreadsheet. The programthen draws a graph of thefunction.



E x a m p l e 4

PHYSICS A football is thrown with the initial velocity of 64 ft/sec. How long does it remain in the air?

SolutionSubstitute 64 for v in the equation. 64t � 16t 2 � 0

Factor the equation. 16t(4 � t) � 0

Set each factor equal to 0 and solve for t. 16t � 0 4 � t � 0

t � 0 t � 4

The equation has two solutions. The first solution represents the launch time; the second represents the landing time. The football stays in the air for 4 sec.

TRY THESE EXERCISES

GRAPHING Use a graphing calculator to determine the number of solutionsfor each equation. For equations with one or two solutions, find the exactsolutions by factoring.

1. 0 � x 2 � 10x � 21

2. 0 � x 2 � 5x � 6

3. 0 � x 2 � 25

4. 0 � x 2 � 14x � 49

5. 0 � x 2 � 18x � 81

6. 0 � x 2 �9


GRAPHING Use a graphing calculator to determine the number of solutionsfor each equation. For equations with one or two solutions, find the exactsolutions by factoring.

7. 0 � x 2 � 100 8. 0 � x 2 � 7x 9. x 2 � 9x � 14 � 0

10. 0 � x 2 � 9x� 25 11. 0 � �x 2 � 4 12. x 2 � x� 0

13. 0 � x 2 � x � 1 14. 0 � x 2 � 10x � 25 15. 0 � x 2 � 8x � 48

Write an equation for each problem. Then factor to solve.

16. The square of a positive integer is 20 less than 12 times the integer. Find the integer.

17. The square of a number exceeds the number by 30. Find the number.

18. The square of an integer is 5 more than 4 times the integer. Find the integer.

19. Seven times an integer plus 8 equals the square of the integer. Find theinteger.


x � �3, x � �7

x � �2, x � �3

no solutions

x � 7

x � �9

x � 3, x � �3

x � 10, x � �10 x � 0, x � �7x � 2, x � 7

x � 0, x � �1

x � 4, x � �12

x � 2, x � �2

x � 5no solution

no solution

10 or 2

6 or �5

5 or �1

8 or �1

For Exercises 20–25, use the equation vt � 16t 2 � 0, where v � velocity in feetper second and t � time in seconds.

20. PHYSICS The initial velocity of a projectile is 128 ft/sec. In how manyseconds will it return to the ground?

21. ARCHERY How long will an arrow shot from ground level with an initialvelocity of 176 ft/sec remain in the air?

22. SPORTS How long will a football kicked with an initial velocity of 96 ft/secremain in the air?

23. PHYSICS A rocket at a fireworks display was launched with the initialvelocity of 208 ft/sec. How many seconds was it in the air before it splasheddown in the lake?

24. PHYSICS A projectile is launched with the initial velocity of 896 ft/sec. How many seconds will it remain in the air?

25. WRITING MATH The equation vt � 16t2 � 0 is of the form y � ax 2 � bx. From Exercises 20–23, what generalization can you make aboutthe solutions to quadratic equations in the form 0 � ax 2 � bx?

Use your generalization from Exercise 25 to solve each of the followingequations.

26. x 2 � 3x � 0 27. x 2 � 2x � 0 28. 5x 2 � 20x � 0

29. 2x 2 � 8x � 0 30. 9x 2 � 72x � 0 31. 3x 2 � 33x � 0


The sum of the first n positive even integers is S � n 2 � n. How many integersmust be added to give each sum?

32. 240 33. 2070 34. 1122 35. 40,200

36. CHAPTER INVESTIGATION Working together, build your gravity-poweredvehicle. Try to keep the weight as close to 10 oz as possible. If necessary addweight by adding new design elements. If your vehicle has wheels, make surethe wheels turn freely. You may want to use a drop of oil or powderedgraphite to lubricate the axle.


Find factors for the following. (Lesson 11-3)

37. 9c � 27b 38. x 2y � x 39. 3mn 2 � 9mn

40. 8a 2b � 32ab 2 41. wx � xz 42. 17a 2b � 68ab

Find the GCF and its paired factor for the following. (Lesson 11-3)

43. 36a � 63b 44. 12ab � 8a 2b

45. 10x 3 � 15x 2 46. 7a 3bc 2 � 14a 2bc

47. 27x 3y 2 � 6x 2y 48. 72p 2q 3r � 32p 2q 4r

49. 6x 3y 3 � 9x 2y 4 � 6x 2y 3z 50. 24a 3b 2c � 18ab 2c 3 � 12abc 2

Lesson 12-3 Factor and Graph 533

8 sec

11 sec

6 sec

13 sec

56 sec

x � 0, x � �2

x � 0, x � 8x � 0, x � �11

x � 0, x � �4x � 0, x � 3

x � 0, x � 4

15 45 33 200

9(c � 3b) x(xy � 1)3mn(n � 3)

17ab(a � 4)x(w � z)8ab(a � 4b)

9(4a � 7b)

5x2(2x � 3)

3x2y(9xy � 2)

3x2y3(2x � 3y � 2z) 6abc(4a2b � 3bc2 � 2c)

8p2q3r (9 � 4q)

7a2bc(ac � 2)

4ab(3 � 2a)

One solution is always 0, the other is ��ba

�.



MODELING Work with a partner to build equations using Algeblocks.

1. Use Algeblocks to illustrate each perfect square.

x 2 � 6x � 9 � (x � 3)2 x 2 � 10x � 25 � (x � 5)2

x 2 � 14x � 49 � (x � 7)2 x 2 � 2x � 1 � (x � 1)2

2. Each of the squares above is in the form x 2 � bx � c � (x � h)2. Discuss the following:

a. What is the relationship between c and h?

b. What is the relationship between h and b?

c. What is the relationship between c and b?

BUILD UNDERSTANDING

Making a perfect square for an expression of the form ax 2 � bx iscalled completing the square. Completing the square is another method for solving quadratic equations.

E x a m p l e 1

Complete the square for x 2 � 8x.

SolutionMODELING Use Algeblocks to illustrate x 2 � 8x. Add blocks tomake a perfect square. Write the expression.

(x 2 � 8x � 16) � 16

In the activity at the top of the page, you discovered that forperfect squares in the form

ax 2 � bx � c, the constant c � ��b2

��2. Thus, by substitution, you

know ax 2 � bx � c � ax 2 � bx � ��b2

��2.

You can use this relationship to complete the square for x 2 � 8x.

x 2 � 8x � ��b2

��2

Add ��b2

��2

to complete the square.

x 2 � 8x � (�4)2 Find �b2

�, ��82

� � �4.

x 2 � 8x � 16 Square �b2

�, (�4)2 � 16.

The expression is (x 2 � 8x � 16) � 16. This is equivalent to theoriginal expression.


12-4 Complete the SquareGoals ■ Solve quadratic equations by completing the square.

Applications Science, Aeronautics, Sports

?

Problem SolvingTip

Use the relationships youdiscover for questions aand b to find c.

c � h2

h � �b2

�

c � ��b2

��2

Animationmathmatters3.com


You can use the process of completing the square to solve quadratic equations.

E x a m p l e 2

Solve by completing the square.

x 2 � 8x � 12 � 0

Solutionx 2 � 8x � 12 � 0

x 2 � 8x � �12 Add �12 to each side.

x 2 � 8x � 16 � �12 � 16 Add ��b2

��2

to each side.

(x � 4)2 � 4 Factor.

x � 4 � �4� Simplify.

x � 4 � �2

x � �4 � 2, x � �4 � (�2)

x � �2, x � �6

The solutions of the equation x 2 � 8x � 12 � 0 are �2 and �6.

To solve a quadratic equation by completing the square, the coefficient of the x 2 term must be 1.

E x a m p l e 3

Mr. Bruno has a square garden in his yard. He wants to double the area by increasing the length 6 ft and the width by 4 ft. Find the original dimensions of the garden.

SolutionLet x be the length of each side of the garden. The original area of the garden is A � x2. He wants this area to be doubled. Write this as a quadratic equation.

2x 2 � (x � 6)(x � 4)

2x 2 � x 2 � 6x � 4x � 24

x 2 � 10x � 24 Simplify.

x 2 � 10x � 25 � 24 � 25 Add ��b2

��2

to each side of the equation.

(x � 5)2 � 49 Factor the left side of the equation.

x � 5 � 7 Take the square root of both sides.

x � 12

The garden was 12 ft on each side.

Lesson 12-4 Complete the Square 535

Problem SolvingTip

Always check yoursolutions by substitutingthem back into theoriginal equation.



TRY THESE EXERCISES

MODELING Complete the square. Use Algeblocks if you wish.

1. x 2 � 4x 2. x 2 � 6x 3. x 2 � 2x

Solve by completing the square. Check your solutions.

4. x 2 � 2x � 8 � 0 5. x 2 � 6x � 7

6. 2x 2 � 5x � 2 � 0 7. x 2 � 4x � 12 � 0

Write an equation. Then complete the square to solve theproblem.

8. The width of a rectangle is 6 cm shorter than the length.The area is 16 cm2. Find the length and width of therectangle.

9. WRITING MATH Explain why you cannot use the negative solution to theequation to find the answer to the problem in Exercise 8.

10. Evan needs to solve the equation x 2 � 2x � 15. After completing the square,he writes x 2 � 2x � 1 � 0, factors the equation as (x � 1)2 � 0 and solves forx. Evan is surprised when his solution, x � 1, doesn’t check. What did he dowrong? Find the correct solutions.


Complete the square.

11. x 2 � 10x 12. x 2 � 20x 13. x 2 � x

14. x 2 � 14x 15. x 2 � 18x 16. x 2 � 30x

17. x 2 � 3x 18. x 2 � 16x 19. x 2 � x


20. x 2 � 3x � 28 � 0 21. 3x 2 � 2x � 5 � 0

22. 9x 2 � 18x � 0 23. x 2 � 3x � 0

24. x 2 � 2x � 1 � 0 25. 2x 2 � 9x � 5

26. x 2 � x � 12 � 0 27. 2x 2 � 5x � 3

28. x 2 � 6x � 7 29. 2x 2 � 4x � 0

30. 6x 2 � 5x � �1 31. x 2 � 6x

Find values for c and h to complete each perfect square.

32. x 2 � 20x � c � (x � h)2 33. x 2 � 4x � c � (x � h)2

34. x 2 � x � c � (x � h)2 35. x 2 � 3x � c � (x � h)2

36. x 2 � 18x � c � (x � h)2 37. x 2 � 22x � c � (x � h)2

38. AERONAUTICS A length of a rectangular panel on a satellite is 4 cm greater than its width. Its area is 165 cm2. Find its dimensions.


A � 16 cm2

x cm

(x � 6) cm

(x2 � 4x � 4) � 4 (x2 � 6x � 9) � 9 (x2 � 2x � 1) � 1

x � �7, x � 1

x � �6, x � 2

x � 4, x � �2

x � 2, x � �12

�

8 cm by 2 cm

lengths cannot be negative

Evan dropped the 15 from the original equation, instead ofsetting the quadratic expression equal to 0. The correctsolutions are x � �3, x � 5.

(x2 � x � �14

�) � �14

�

(x2 � 30x � 225) � 225

(x2 � x � �14

�) � �14

�

(x2 � 20x � 100) � 100(x2 � 18x � 81) � 81

(x2 � 16x � 64) � 64

(x2 � 10x � 25) � 25

(x2 � 14x � 49) � 49

(x2 � 3x � �94

�) � �94

�

x � �4, x � 7

x � 0, x � 2

x � 1

x � 4, x � �3

x � �1, x � 7

x � �12

�, x � �13

� x � 0, x � 6

x � �1, x � �53

�

x � �3, x � 0

x � ��12

�, x � 5

x � �3, x � �12

�

x � 0, x � 2

c � 4, h � �2

35. c � �94

�, h � � �32

�

c � 121, h � 11

11 cm � 15 cm

c � 100, h � 10

c � ��14

�, h � �12

�

c � 81, h � �9

39. PHYSICS A ball is thrown from the ground up into the air at an initialvelocity of 64 ft/sec. How long will it take the ball to reach the ground? Usethe equation 64t � 16t 2 � 0, where t equals time in seconds.

40. SPORTS A length of a rectangular playing field is twice the width plus 4 ft.Its area is 2310 ft2. Find the length, width, and perimeter in feet of the field’sboundary line.


Solve each problem. If necessary, write an equation and then complete thesquare.

41. The width of a rectangle is 2 in. less than its length. Find its dimensions if itsarea is 360 in.2.

42. If the width and length of a 4-in. by 2-in. rectangle are each increased by thesame amount, the area of the rectangle will be 48 in.2. Find the new lengthand new width.

43. A triangle with an area of 6 m2 has a base that is 4 m longer than its height.What are the dimensions?

44. If the height and base of a triangle with a height of 5 cm and a base of 8 cmare each decreased by the same amount, the area of the triangle will be 14 cm2. Find the new base and height.

45. ART A painting is 1 in. longer than it is wide. The painting and its framehave a total area of 156 in.2. The frame is 1 in. wide on each side of thepainting. What are the dimensions of the painting?

46. CHAPTER INVESTIGATION Work with your group to build a ramp from24–36 in. in length. One possibility would be to attach a length of posterboard to two yard sticks, using the sticks for stability. Set the ramp up so thatthe angle formed by the ramp and floor is 15°. Using a stopwatch, time yourvehicle’s descent from the top of the ramp to the bottom. Make sure you donot push the vehicle at the starting point. Increase the ramp’s incline in 5°increments. How does the increase affect your vehicle’s travel time? For eachangle of descent, measure the distance the vehicle travels beyond the bottomof the ramp. How does changing the incline affect this distance?



47. (3a � 2b)(2a � 4) 48. (6x � 2)(4k � 3) 49. (2a � 3c)(4b � 2d)

50. (5x � y)(3x � 1) 51. (3c � 2)(3c � 2) 52. (5a � 1)(5a � 1)

53. (4x � 2)(3x � 1) 54. (x � 9)(2x � 4) 55. (7a � 2b)(5a � 3b)

56. (2c � 5b)(4c � 3b) 57. (12x � 2y)(5x � 6y) 58. (4b � 7y)(4b � 7y)

Write each in simplest radical form. (Lesson 10-1)

59. �77� 60. �112� 61. �56�

62. (2�5�)(3�10�) 63. (2�17�)(4�22�) 64. (5�11�)(4�32�)

65. 66. 67. ��81

02��40�

��5�

�18��6�

Lesson 12-4 Complete the Square 537

4 sec

18 in. � 20 in.

8 in., 6 in.

h � 2 m, b � 6 m

b � 7 cm, h � 4 cm

10 in. � 11 in.

6a2 � 12a � 4ab � 8b

15x2 � 5x � 3xy � y

12x2 � 2x � 2

8c2 � 26bc � 15b2 60x2 � 62xy � 12y2

2x2 � 14x � 36

24xk � 18x � 8k � 6

9c2 � 12c � 4

8ab � 4ad � 12bc � 6cd

25a2 � 1

35a2 � 11ab � 6b2

16b2 � 49y2

�77�

30�2�

�3� 2�2�

8�374�

4�7� 2�14�

80�22�

�2�

315��

width � 33 ft, length � 70 ft, perimeter � 206 ft



PRACTICE LESSON 12-3Determine if each statement is true or false.

1. There are always 2 solutions to every quadratic equation.

2. The x-coordinate of a point at which the graph of a quadratic functionintersects the x-axis is a solution of the related quadratic equation.

3. For every quadratic equation, you can find exact solutions of an equation by factoring the equation, then substituting the value of x in the equation to find y.

4. If a parabola graphed on the coordinate plane does not meet the x-axis, howmany real solutions of the related quadratic equation are there?

Use a graphing calculator to determine the number of solutions for eachequation. For equations with one or two solutions, find the exact solutions byfactoring.

5. 0 � x 2 � x � 12 6. 0 � x 2 � 6x � 8 7. 0 � x 2 � 4x � 4

8. 0 � x 2 � x � 6 9. 0 � x 2 � 9x � 25 10. 0 � x 2 � 7x � 10

Write an equation for each problem. Then factor to solve.

11. The square of a positive integer is 20 more than the integer.

12. The square of a positive integer is 9 less than 6 times the integer.



PRACTICE LESSON 12-415. For a perfect square in the form ax 2 � bx � c, what formula can be

substituted for the constant c?

16. To solve a quadratic equation by completing the square, what must be trueabout the x 2 term?


17. x 2 � 4x 18. x 2 � 10x 19. x 2 � 18x

20. x 2 � 14x 21. x 2 � 5x 22. x 2 � 13x


23. 3x 2 � 5x � 2 � 0 24. x 2 � x � 2 25. x 2 � x � 0

26. 2x 2 � 7x � 15 � 0 27. x 2 � 4x � 12 28. 8x 2 � 6x � �1

29. The length of a rectangle is 4 in. greater than its width. Find its dimensions ifits area is 32 in.2.

30. The length of a swimming pool is 3 times its width. Find its dimensions if itsarea is 432 ft2.


false

true

false

x � �3, 4

x � �2, 3

x � �2, �4x � �2

x � 2, 5no solutions

y � x2 � x � 20; x � 5

y � x2 � 6x � 9; x � 3

y � x2 � 3x � 2; x � 1, 2

y � x2 � 5x � 6; x � 2, 3

c � ��b2

��2

It must have a coefficient of 1.

(x2 � 4x � 4) � 4

(x2 � 14x � 49) � 49

(x2 � 10x � 25) � 25(x2 � 18x � 81) � 81

(x2 � 13x � �1649

�) � �16

49

�(x2 � 5x � �

245�) � �

245�

�1, 2

�6, 2 �14

�, �12

�

w � 4 in., � � 8 in.

��13

�, 2

��32

�, 5

w � 12 ft, � � 36 ft

0, 8

no solutions


PRACTICE LESSON 12-1–LESSON 12-4Copy and complete each table. Then draw the graph. (Lesson 12-1)

31. y � x 2 � 5 32. y � 3x 2 � 2

Find the vertex and axis of symmetry. (Lesson 12-2)

33. y � x 2 � 3x � 4 34. y � 3x 2 � x � 3 35. y � �2x 2 � 4x � 1

36. y � �x 2 � 5x � 3 37. y � 2x 2 � 3x � 4 38. y � �x 2 � 4x � 7

Solve each equation by factoring or by completing the square.(Lessons 12-3–12-4)

39. y � x 2 � 12x � 1 40. y � x 2 � 4x � 3 41. y � x 2 � x � 12

42. y � x 2 � 5x � 14 43. y � x 2 � 10x � 4 44. y � x 2 � 2x � 15

45. y � x 2 � 8x � 12 46. y � x 2 � 7x � 2 47. y � x 2 � 6x � 7

48. The length of a rectangle is 7 in. greater than its width. Find its dimensions ifits area is 144 in.2.

Mid-Chapter QuizGraph each function. Then name the vertex and the axis of symmetry, and statewhether the graph opens upward or downward. (Lesson 12-1)

1. y � 7x 2 2. y � �3x 2 � 1

Use x � ��2ba� to find the coordinates of the vertex for each parabola. (Lesson 12-2)

3. y � 2x 2 � x � 1 4. y � �3x 2 � 2x � 2

Write the equation for the line of symmetry for the graph of each equation.(Lesson 12-2)

5. y � 2x � x 2 � 4 6. y � 3 � 3x 2 � 6x

Find the solution or solutions of each equation by factoring. (Lesson 12-3)

7. 0 � x 2 � 2x 8. 0 � �x 2 � 4x � 4

9. 0 � �x 2 � 3x 10. 0 � x 2 � 3x � 4

Solve by completing the square. (Lesson 12-4)

11. x 2 � 4x � 3 � 0 12. 2x 2 � 16x � 18 � 0

13. x 2 � x � 3�34

� 14. x 2 � 20x � 21

x �3 �2 �1 0 1 2 3y

x �3 �2 �1 0 1 2 3y



��32

�, �74

��; x � �32

�

��16

�, �3152��; x � ��

16

�

(�2, �3); x � �2

(�1, 3); x � �1

��52

�, �143��; x � �

52

�

x � ��37� � 6

x � �2, 7

x � �6, �2

w � 9; � � 16

x � 1, 3

x � ��29� � 5

x � ��14.25� � 3.5

x � �4, 3

x � �3, 5

x � �1, 7

(0, 0); x � 0; upward


(0, 1); x � 0; downward

��13

�, 2�13

��14

�, �1�18

��

x � 1 x � �1

�2

�4, 1

x � 1, x � �9

x � �1, x � 21

x � 1, x � 3

x � �1�12

�, x � 2�12

�

�2, 0

�3, 0

��34

�, �283��; x � �

34

�

Work in groups of 2–3 students.

By looking for relationships among the variables, coefficients and constants inequations, we can write general rules or steps for solving all equations of thesame form.

1. Work together to write a list of steps for solving any quadratic equation in theform ax 2 � bx � c � 0 by completing the square.

2. Check your steps by using them to solve the equation x 2 � x � 3�34

�.

BUILD UNDERSTANDING

Any quadratic equation can be solved by completing the square; however,repeating the steps from the activity above for each equation you solve can be alengthy process. Instead of repeating the steps, you can use the general quadraticequation ax 2 � bx � c � 0 to develop a formula for solving quadratic equations.The formula is found by solving the general quadratic equation for x.

ax 2 � bx � c � 0

x 2 � �ab

�x � �ac

� � 0 Multiply each term by �a1

�.

x 2 � �ab

�x � ��ac

� Add ��ac

� to each side of the equation.

x 2 � �ab

�x � � � �ac

� Half of �ab

� � �2ba�. Add ��

2ba��

2or �

4ba

2

2�

to each side of the equation.

�x � �2ba��

2� Factor the left side of the equation.

Combine terms on the right side of the equation.

x � �2ba� � � Find the square roots.

x � ��2ba� � �� Subtract �

2ba� from each side of the equation.

x � � Simplify.

The formula for x in terms of a, b, and c is called the quadratic formula.

x �

The quadratic formula can be used to solve any quadratic equation of the form ax 2 � bx � c � 0, a � 0.

�b � �b 2 � 4�ac��

2a

b � �b 2 � 4�ac��

2a

b 2 � 4ac��

2a

��b2

4�

a2

4�ac��

b 2 � 4ac��

4a 2

b 2

�4a 2

b 2

�4a 2

12-5 The Quadratic FormulaGoals ■ Solve equations using the quadratic formula.

Applications Aeronautics, Skydiving, Physics


1. Add �c to each side. Multiply by �1a

�. Add

to each side. Factor. Find square roots.

Simplify.��b2

��2x � �1�

12

�, x � 2�12

�

E x a m p l e 1

Use the quadratic formula to solve 6x 2 � 5x � 1 � 0.

Solutionx �

x � Substitute for a, b, and c.

x � Simplify.

x � �5 �

12�1��

51�

21

�

x � �5

1�

21

� and x � �5

1�

21

�

x � �162� or �

12

� x � �142� or �

13

�

The equation 6x 2 � 5x � 1 � 0 has two solutions, �12

� and �13

�.

The radical part of the solutions is in simplified form if it contains no factors thatare perfect squares.

E x a m p l e 2

Use the quadratic formula to solve x 2 � 6x � 3.

Solution�x 2 � 6x � 3 � 0 Write the equation in standard form.

�x � Use the quadratic formula.

�x � Substitute for a, b, and c.

�x � Simplify.

�x � �6 �

2�48��

�x � �6 �

24�3�� or 3 � 2�3�

The solutions for the quadratic equation x 2 � 6x � 3 are 3 � 2�3� and 3 � 2�3�.

You can use your knowledge of quadratic equations to solve many problemsinvolving distance. For example, gravity acts on a freely falling object accordingto the formula h � 16t 2. Using this formula, you can find the time it takes for anobject to fall a certain distance.

Now consider the path of a projectile. Velocity, the force applied to the object, and gravity act upon the object. The path of a projectile is described bythe formula h � vt � 16t 2, where h � height (ft), v � velocity (ft/sec) and t � time (sec).

6 � �36 � 1�2��

2

�(�6) � �(�6)2�� 4(1)�(�3)��

2(1)

�b � �b 2 � 4�ac��

2a

5 � �25 � 2�4��

12

�(�5) � �(�5)2�� 4(6)�(1)��

2(6)

�b � �b2 � 4�ac��

2a

Lesson 12-5 The Quadratic Formula 541

Problem SolvingTip

Remember the productproperty of square roots:

For any non-negative realnumbers a and b,

�a� � �b� � �ab�

so, �12� � �4 � 3� �

�4� � �3� � 2�3�





E x a m p l e 3

AERONAUTICS A model rocket is launched at a velocity of 80 ft/sec.

a. How long does it take the rocket to reach its maximum height?

b. What is the maximum height reached by the rocket?

c. How many seconds does it take the rocket to return to the ground?

SolutionThe velocity is 80 ft/sec, so h � 80t � 16t 2.

a. At its maximum height, the rocket is at the vertex of the parabola formed by the height function. The time required to reach the maximum height is the

x-coordinate of the vertex. Use ��2ba� to find the time.

��2(�

8016)� � �

83

02� � 2.5

It takes the rocket 2.5 sec to reach its maximum height.

b. The maximum height is the y-coordinate of the vertex.Substitute 2.5 for t in the equation.

h � 80(2.5) � 16(2.5)2

� 200 � 16(6.25)

� 200 � 100 � 100

The maximum height reached by the rocket is 100 ft.

c. At ground level, h is 0. To find when the rocket returns to theground, solve the related quadratic equation.

80t � 16t 2 � 0

5t � t 2 � 0

t(5 � t) � 0 t � 0, t � 5

The rocket returns to ground level 5 sec after launch.

TRY THESE EXERCISES

Use the quadratic formula. Solve each equation.

1. x 2 � 4x � 1 � 0 2. 2x 2 � 15x � 8 3. 4x 2 � 8x

4. �6x � �3x 2 � 3 5. x 2 � 2x � 6 6. 7x � 15 � �2x 2


Use the quadratic formula to solve each equation.

7. x 2 � 9x � �14 8. 2x 2 � 6 � x 9. 4x 2 � 4x � 1 � 0

10. x 2 � 2x � 11 11. 14x � 2x 2 12. 3x 2 � 84 � 9x

13. 2x 2 � 32 � 16x 14. x 2 � 50 15. x 2 � 4x � 2 � 0

16. x 2 � 7x � �12 17. 45 � 2x 2 � x 18. x 2 � 8 � 0


120

100

80

60

40

20

0 1 2 3 4 5

Heightin feet

Time in seconds

2 ��3�

1

�12

�, �8

1 ��7�

0, 2

1.5, �5

2, 7

�1 � 2�3�

4

�3, �4 4�12

�, �5

�32

�, �2

0, 7

�5�2�

��12

�

4, �7

�2 ��2�

�2�2�

Choose factoring or the quadratic formula to solve each equation.

19. x 2 � 2x � 35 20. 2x � 4 � �x 2 21. x 2 � �3x

22. x 2 � 2x � 7 23. x � 21 � �2x 2 24. x � x 2

25. 6x 2 � 2 � x 26. x 2 � 10x � 3 27. x 2 � 12x �11

28. WRITING MATH Suppose Galileo gathered thedata shown. Explain how Galileo might use this data to show that free-fall distance is a function of the square of time.

29. SKYDIVING A parachutist free-falls for 20 sec. How far is the free-fall?

30. PHYSICS How many times longer does aprojectile launched at 200 ft/sec stay in the air than one launched at 100 ft/sec?


31. AERONAUTICS A model rocket is launched at an initialvelocity of 96 ft/sec.

a. How long will it take the rocket to reach its maximumheight?

b. What is the maximum height reached by the rocket?

c. How many seconds does it take the rocket to return to theground?

32. PHYSICS An object free-falls for 15 sec, and another free-falls for 30 sec. How many times farther does the second object fall than the first?

33. DATA FILE Use the data on the heights of some bridges on page 644. Howlong would it take a stone dropped from each bridge to reach the waterbelow? Find your answer to the nearest tenth of a second.

34. PHYSICS A ball is thrown up into the air from the roof of a 128-ft-tallbuilding. The initial velocity is 64 ft/sec. How long will it take the ball toreach the ground?


Find factors for the following. (Lesson 11-5)

35. 2a 2 � 5ab � 3b 2 36. 18x 2 � 3xy � 6y 2 37. 16a 2 � 16ab � 4b 2

38. 8x 2 � 18xy � 9y 2 39. 6m 2 � 7mn � 20n 2 40. 36a 2 � 12ab � 8b 2

41. 24a 3 � 16a 2b � 8ab 2 42. 10z 2 � 26xz � 12x 2 43. 8m 2 � 10mn � 12n 2

44. 49a 2 � 28ab � 4b 2 45. 6x 2 � 4xy � 2y 2 46. 3a 2 � 2ab � 5b 2

Solve each system of equations. (Lesson 6-6)

47. 3x � 2y � �14 48. 3y � x � 104x � 2y � 0 y � 2x � 1

Lesson 12-5 The Quadratic Formula 543

Math: Who,Where, When

Italian physicist andastronomer Galileo Galilei(1564–1642) probably didnot drop cannonballsfrom the Leaning Towerof Pisa. However, from hisresearch came thequadratic law of fallingbodies.

5, �7�1 � �5� 0, �3

0, 1

1, 115 � 2�7�

��72

�, 31 � 2�2�

��23

�, �12

�

The distance is equal to a constant (16) times the time squared.

6400 ft

2 times

3 sec

144 ft

6 sec

4 times

about 5.5 sec

(a � b)(2a � 3b) 3(3x � 2y)(2x � y)

(3m � 4n)(2m � 5n)

2(z � 3x)(5z � 2x)

2(3x � y)(x � y)

(�2, 4) (1, 3)

4(2a � b)(2a � b)

4(3a � b)(3a � 2b)

2(m � 2n)(4m � 3n)

(3a � 5b)(a � b)

(4x � 3y)(2x � 3y)

8a(3a � b)(a � b)

(7a � 2b)(7a � 2b)

Firth of Forth Bridge, 3.0 sec;Verrazzano Narrows Bridge, 3.6 sec; Sydney Harbor Bridge, 3.3 sec; Tunkhannock Viaduct 3.9 sec; Garabit

Viaduct, 5.5 sec;Brooklyn Bridge, 2.9 sec.

16 � 16(12)64 � 16(22)144 � 16(32)256 � 16(42)


Time(seconds)

Distance(feet)

0 01 162 643 1444 256


Work with a partner.

1. Using the coordinate plane shown below, count squares to find the length andmidpoint of each leg of the right triangle.

2. Note that B�C�is parallel to the y-axis, and A�C�is parallel to the x-axis. Answerthe following questions.

a. How can you calculate the length of B�C� given the y-coordinates of B and C ?

b. The midpoint of B�C�is halfway between the endpoints B and C. How canyou calculate the y-coordinate of the midpoint, given the y-coordinates ofB and C ?

c. How can you calculate the length of A�C�, given the x-coordinates of A and C ?

d. How can you calculate the x-coordinate of the midpoint of A�C�?

3. Use the Pythagorean Theorem to find the length of hypotenuse A�B�.

4. The midpoint of A�B�is (4, 5). What is the relationship between this point andthe midpoints of the two legs?

BUILD UNDERSTANDING

A formula for calculating the distance, d, between any two points on the coordinate plane may be derived using the Pythagorean Theorem.

For any two points (x1, y1) and (x2, y2), a right triangleformed by drawing horizontal and vertical segmentsthat intersect at (x2, y1) has legs with lengths �x2 � x1�and �y2 � y1�.

These lengths may be substituted in the PythagoreanTheorem to find d.

d � �(x2 ��x1)2 �� (y2 �� y1)2�

544

12-6 The Distance FormulaGoals ■ Use the Pythagorean Theorem, distance and mid-

point formulas.

Applications Space Exploration, Sports, Archaeology


y

x

9

8

7

6

5

4

3

2

1

1 2 3 4 5 6 7 8 9�1 10A(1,1) C(7,1)

B(7,9)

y

x

7

6

5

4

3

2

1

1 2 3 4 5 6 7 8 9�1 10

(x1,y1) (x2,y1)

(x2,y2)

x2 � x1

y2 � y1

Problem SolvingTip

From the Pythagorean Theorem, c2 � a2 � b2.Solving for c,

c � �a2 � b�2�.

AC:6, (4, 1);BC:8, (7, 5)




10


The distance, d, between any two points (x1, y1) and (x2, y2) may be found usingthe distance formula.

d � �(x2 ��x1)2 �� (y2 �� y1)2�

E x a m p l e 1

SPACE EXPLORATION Suppose the grid shown was superimposedover a photograph taken by a space probe of the Martian landscape.Calculate the distance between points P(�2, 4) and Q(4, �1).

Solutiond � �(x2 ��x1)2 �� (y2 �� y1)2�

d � �(�2 �� 4)2 �� (4 � (��1))2�

d � �(�6)2�� 52�

d � �36 � 2�5� � �61� � 7.8

The distance between points P and Q is �61� or about 7.8 units.

In the activity at the beginning of the lesson, you observed that for a segment

parallel to the x-axis, the x-coordinate of the midpoint is �x1 �

2x2

�. For a

segment parallel to the y-axis, the y-coordinate of the midpoint is �y1 �

2y2

�.

We can use these two facts to derive the midpoint formula. For a line segment with endpoints (x1, y1) and (x2, y2), the coordinates of the midpoint are

��x1 �

2x2

�, �y1 �

2y2

��.

E x a m p l e 2

Find the midpoint of the line segment with endpoints M(�2, 5) and N(6, �3).

Solution

��x1 �

2x2

�, �y1 �

2y 2

�� Use the midpoint formula.

��22� 6�, �5 �

2(�3)�� Substitute coordinate values.

��42

�, �22

�� (2, 1) Simplify.

The midpoint of line segment MN is (2, 1).

Distances on aerial and satellite photos are often estimated using a coordinategrid. Even sporting events require these calculations. Suppose you are workingfor a television company broadcasting a major league baseball game. A blimp istransmitting overhead shots of the field at all times. Using a computer, you couldinstantly overlay each visual with a grid and calculate distances for theannouncers to use in the broadcast.

Lesson 12-6 The Distance Formula 545

y

x

4

2

�2

2 4�2

P

Q





E x a m p l e 3

SPORTS On the grid shown at the right, one unitrepresents 30 ft. A batter hits a ball from home plate topoint L. What is the distance?

SolutionThe coordinates of home plate are (�6, �5). Point L islocated at approximately (�4, 5). Use the distanceformula to find the distance between the points.

d � �(x2 ��x1)2 �� (y2 �� y1)2�

d � �(�4 �� (�6))�2 � (5� � (��5))2�

d � �22 � 1�02�

d � �4 � 10�0� � �104� � 10.2 units

Since each unit equals 30 ft, multiply by 30: 10.2 � 30 � 306 ft. The ball traveledabout 306 ft.

TRY THESE EXERCISES

Use the graph on the right. Round distancesto the nearest tenth of a unit.

1. Find the length of segment AD.

2. Find the distance between B and C.

3. Locate the midpoint of segment JK.

Use the distance formula to calculate the distance between each pair of points.Round to the nearest tenth.

4. G(�9, 4), H(�5, �1) 5. P(3, �2), Q(�2, 11) 6. M(0, 0), N(6, �7)

Use the midpoint formula to find the midpoint of the segment with the given endpoints.

7. E(5, 5), F(�3,�3) 8. L(�10, 4), M(2, 6) 9. T(�9, 4), S(1, �1)

10. SPORTS Use the baseball field diagram from Example 3. One unit on thegrid equals 30 ft. If a ball is caught at point H and thrown to second base atcoordinates (�3, �2), how long is the throw?


Calculate the distance between each pair of points. Round to the nearest tenth.

11. W(20, �1), X(�6, 5) 12. C(8, 5), D(�3, �6)

13. G(�1, 5), H(2, 6) 14. P(5, 5), Q(�6, 1)


6

4

2

�2

�6

y

x2�2 4�6 6

A

D

B

C

J

K

6

4

2

�2

�4

�6

2 4�2�4 6

y

x

A

L

T

F

P

C

H

B

7.1 units

12.2 units

(1, 0)

6.4 units 13.9 units 9.2 units

(�4, 1.5)(�4, 5)(1, 1)

about 234 ft

26.7 units

3.2 units

15.6 units

11.7 units

Find the midpoint of the segment with the given endpoints.

15. E(16, �5), F(4, 1) 16. Y(�10, �9), Z(�2, �3)

17. ARCHAEOLOGY The diagram of an archaeological site isdrawn on a coordinate grid. An archaeologist notes thatstanding stones are placed at A(�3, �1), B(1, 3), and C(7, �3). The stones are connected by a low wall to form a triangle. What are the coordinates of the midpoint of thesegment connecting the midpoints of A�B� and A�C� ?

18. What type of triangle is formed by connecting themidpoints of line segments formed by L(�1, �2), M(�1, 4), N(6, �2)?

SPORTS Use the baseball field diagram from Example 3 for Exercises 19–22.Estimate distances on the ground. One unit on the grid represents 30 ft.

19. A home run is hit from home plate to point B. What is the distance?

20. A foul ball is hit from home plate to point F. What is the distance?

21. The right fielder catches a ball at point C and throws the ball to first base atthe point with coordinates (�3, �5). How long is the throw?

22. What is the home-run distance from home plate to point A?

23. DATA FILE Use the diagram of a soccer field on page 653. Using only thelengths given and a grid overlay, estimate the following distances shown bycolored arrows: green (Neyome to Young), red (Young to Lato), purple(Kasberczak to Carr), and black (Young to Correa).

24. YOU MAKE THE CALL The point (3, 5) is the midpoint of a segment thathas (7, 11) as one endpoint. Janice says there isn’t enough information tofind the other endpoint of the segment. Do you agree? Explain your thinking.

25. WRITING MATHEMATICS When finding the distance between two points(1, 4) and (8, 3), explain why it makes no difference which point you use as(x1, y1) and (x2, y2).


26. Use the distance formula to find the equation for a circle with radius 5 andcenter at point (0, 0).

27. Use the distance formula to find the equation for a circle with radius r andcenter at point (0, 0).

28. Find an equation for the circle with center (0, 2) and radius 8.


Find binomial factors for the following, if possible. (Lesson 11-7)

29. 3x 2 � 10x � 8 30. 2x 2 � 3x � 5 31. 6x 2 � 26x � 24

32. x 2 � 4x � 4 33. 2x 2 � 13x � 24 34. 15x 2 � x � 2

35. 12x 2 � 14xy � 6y 2 36. 28x 2 � 30xy � 8y 2 37. 10x 2 � 41x � 21

38. 24x 2 � 4xy � 8y 2 39. 16x 2 � 24xy � 9y 2 40. 2x 2 � 16xy � 30y 2

Lesson 12-6 The Distance Formula 547

Mesa Verde National Park CliffDwellings, Colorado

(10, �2)(�6, �6)

(0.5, �0.5)

a scalene right triangle

about 467 ft

about 124 ft

about 234 ft

about 381 ft

green: 18 yd, red: 58 yd, purple: 18 yd;black: 38 yd

Janice is wrong. If the coordinates of the other endpoint are (x, y), then (3, 5) � ��7 �2

x�, �112� y��.

Solve for x and y to find the other endpoint.

Answers will vary. Since (x2 � x1)2 � (x1 � x2)2 and(y2 � y1)2 � (y1 � y2)2, the sum of the squared differences will be the samepositive number.

5 � �(x � 0�)2 � (�y � 0)�2� � 5 � �x2 � y�2� � 25 � x2 � y2

x2 � y2 � r2

(x � 0)2 � (y � 2)2 � 64, x2 � y2 � 4y � 4 � 64, x2 � y2 � 4y � 60

(x � 4)(3x � 2)

(x � 2)(x � 2)

2(3x � y)(2x � 3y)

4(3x � 2y)(2x � y)

(2x � 5)(x � 1)

not factorable

2(2x � y)(7x � 4y)

(4x � 3y)(4x � 3y)

2(x � 3)(3x � 4)

(5x � 2)(3x � 1)

(2x � 7)(5x � 3)

2(x � 5y)(x � 3y)



PRACTICE LESSON 12-5Determine if each statement is true or false.

1. The quadratic formula was derived from the standard form of a quadraticequation.

2. The quadratic formula cannot be used if there is no x term in the quadratic equation.

3. The equation 0x2 � 5x � 3 � 0 is not a quadratic equation.

4. The quadratic formula cannot be used to solve 0x2 � 8x � 4 � 0. Explain why.


5. x 2 � 7x � 2 � 0 6. 2x 2 � 5x � 3 � 0 7. x 2 � 4x � 3 � 0

8. 3x 2 � 4x � 1 � 0 9. �2x 2 � 3x � 5 � 0 10. �x 2 � 4x � 8 � 0

11. 2x 2 � 7x � 2 � 0 12. 4x 2 � x � 2 � 0 13. �2x 2 � x � 5 � 0

PRACTICE LESSON 12-614. When finding the distance on a coordinate plane, either

endpoint may be designated as (x1, y1), or (x2, y2). Give an example that proves or disproves this theory.

Use the graph on the right. Round distances to the nearesttenth of a unit.

15. Find the length of segment AB.

16. Find the distance between C and D.

17. Find the midpoint of segment EF.


18. Y(1, 6), Z(�4, 3) 19. W(2, �5), X(4, 2) 20. U(1, �5), V(3, 7)

21. S(�4, �2), T(�5, 3) 22. Q(7, �2), R(�5, 3) 23. O(3, 3), P(�3, �3)

Calculate the midpoint of the segment with the given endpoints.

24. A(�4, 3), B(3, �4) 25. C(1, 5), D(3, �2) 26. E(4, �7), F(�2, 3)

27. G(5, 3), H(�5, �3) 28. I(6, �1), J(�2, 5) 29. K(4, 6), L(1, �3)

Calculate the midpoint and distance between each pair of points.

30. Z(�3, 5), Y(�2, �6) 31. X(4, �5), W(�2, 1) 32. V(0, 3), U(5, 4)

33. T(�1, 4), S(�2, �3) 34. R(2, 2), Q(�5, �5) 35. P(4, �8), O(�8, 4)

36. Use the distance formula to find an equation for the circle with a radius of 4and center at point (0, 0).


y

x�5 5

5

�5

EA

F

BC

D

true

false

true

The equation is not a quadratic equation. The value of the denominator would be 0.

x � �7 �

2�41��

x � 1, �13

�

x � �7 �

4�33��

7.8

12.0

��12

�, �72

��

5.8

5.1

7.3

13

12.2

8.5

(1, �2)

��52

�, �32

��

m � ��52

�, �72

��, d � 5.1

m � (�2, �2), d � 17.0m � ��32

�, ��32

��, d � 9.9

m � (1, �2), d � 8.5

(2, 2)

�2, �32

��12

�, ��12

��(0, 0)

m � ��52

�, ��12

��, d � 11.0

m � ��32

�, �12

��, d � 7.1

x2 � y2 � 16

x � �3, �12

�

x � 1, ��52

�

x � ��1 �

8�33��

x � �1 �

4�41��

x � 3, 1

x � 2 � 2�3�

Check students’examples.Because thedifferences aresquared, eitherpoint may be (x1, y1) or (x2, y2).


Workplace Knowhow

PRACTICE LESSON 12-1–LESSON 12-6For each equation, find the vertex and axis of symmetry of the graph. (Lessons 12-1–12-2)

37. y � x 2 � 3x � 2 38. y � �2x 2 � x � 2 39. y � 3x 2 � 4x � 2

40. y � �x 2 � 5x � 4 41. y � 2x 2 � 5x � 1 42. y � x 2 � 7x � 3

Solve each equation by graphing, factoring, or completing the square.(Lessons 12-3–12-4)

43. y � x 2 � 5 44. x 2 � x � 6 � 0 45. x 2 � 6x � 8 � 0

46. x 2 � 6x � 2 � 0 47. x 2 � 8x � 15 � 0 48. y � 2x 2 � 4

49. x 2 � 3x � 4 � 0 50. x 2 � 15x � 56 � 0 51. x 2 � 5x � 1 � 0

52. The width of a rectangular mirror is 9 in. less than its length. Find itsdimensions if the area of the mirror is 360 in.2.

Solve each equation using the quadratic formula. (Lesson 12-5)

53. x 2 � 4x � 3 � 0 54. �2x 2 � 5x � 2 � 0 55. 2x 2 � 4x � 9 � 0


A ir traffic controllers manage the movement of air traffic through sections ofair space. Controllers work together to monitor the movement of an aircraft

from one section to another. They keep aircraft a safe distance apart and work to keep departures and arrivals on schedule. Controllers use radar and visualobservation to monitor the progress of all aircraft. They also monitor the weatherconditions for pilots. Air traffic controllers communicate directly with the pilot todirect the path of the flight. Together, controllers and flight crews make coursecorrections and respond to dangers caused by other aircraft, weather oremergency situations on the ground.

1. To determine the distance between two planes in the air, a three-coordinatesystem must be employed using ordered triplets (x, y, z). Each plane’s position ismeasured in relation not only to a horizontal x-axis and a vertical y-axis, but alsoto a depth-measuring z-axis perpendicular to the other two axes. Place the origin(0, 0, 0) of the three axes at the O’Hare Airport control tower and measure units inmiles. Using the axes and units, describe a plane with coordinates (8, 2, 5).

2. The space distance d between two points (x1, y1, z1) and (x2, y2, z 2) is given by d � �(x2 ��x1)2 �� (y2 �� y1)2 �� (z2 �� z1)2�. Two planes have positioncoordinates (�3, 5, �4) and (9, 2, 0). How far apart are they?

3. A passenger jet is midway between two private aircraft with position coordinates(6, 2, �13) and (1, �8, �5). Find the coordinates of the jet. Explain how youfound your answer.

��32

�, ��147��;

x � ��32

��52

�, �441��;

x � ��52

�

��23

�, �23

��;x � �

23

�

x � 2, �4

no solution

x � ��5 �

2�29��x � 7, 8

x � 3, 5

x � �2, 3x � ��5�

x � 3 � �7�

x � 1, �4

w � 15, � � 24

x � 2 � �7� x � ��5 �

4�41�� x � �

�2 �

2�22��

8 mi along x-axis, 2 mi on the y-axis and 5 mi on the z-axis

about 13 mi

��14

�, ��185��;

x � �14

�

��54

�, ��383��; x � �

54

� ��72

�, ��641��; x � ��

72

�

Career – Air Traffic Controllers

mathmatters3.com/mathworks

��72

�, �3, �9�; I found the average of the x-coordinates, the average of they-coordinates, and the average of the z-coordinates.

http://mathmatters3.com/mathworks

You have learned that the shape of the graph of a quadratic function isa parabola. You also know that quadratic functions are written in theform y � ax 2 � bx � c.

Suppose you were given the graph of a parabola. Can you use yourknowledge to work backwards to discover the equation of the graph?

P r o b l e m

SCIENCE The results of anexperiment are represented by theparabola shown at the right. Whatis the equation of the graph?

Solve the ProblemYou can determine the equation ofa parabola if you know three pointson its graph.

Begin with the general form of a quadratic function: y � ax 2 � bx � c, where a, band c represent coefficients and constants.

Locate three points on the graph. As you can see, the parabola passes through (0, �3), (�1, �4) and (�3, 0).

Substitute the x-values and y-values for each point into the equation to create asystem of three equations.

For (0, �3) For (�3, 0) For (�1, �4)ax 2 � bx � c � y ax 2 � bx � c � y ax 2 � bx � c � yc � �3 9a � 3b � c � 0 a � b � c � �4

Now use any of the methods you have learned c � �3for solving systems of equations. 9a � 3b � c � 0

a � b � c � �4

Substitute �3 for c in the other two equations. Then solve for a.

9a � 3b � 3 � 0 � 9a � 3b � 3 � 0a � b � 3 � �4 � 3a � 3b � 9 � �12 Multiply by 3.

6a � 6 � 12 Subtract

Since 6a � 6 � 12, a � 1. Substitute 1 for a and �3 for c in the second equation.9a � 3b � c � 0

9 � 3b � 3 � 0

6 � 3b

b � 2


12-7 Problem Solving Skills:Graphs to Equations

y

x

4

2

�2

�4

2 4 �4 �2

Problem SolvingStrategies

Guess and check

Look for a pattern

Solve a simplerproblem

Make a table, chartor list

Use a picture,diagram or model

Act it out

Work backwards

Eliminate possibilities

Use an equation orformula

✔

Thus, a � 1, b � 2 and c � �3. Check these values in the third equation. Then use these values in the form y � ax2 � bx � c to write an equation.

The equation of the parabola is y � x 2 � 2x � 3.

TRY THESE EXERCISES

1. The graph of a quadratic function contains the three points (�5, 10), (0, �5) and (2, 3). Find the equation of the function.

2. The graph of a quadratic function contains the three points (2, 7), (0, �1) and (�1, 1). Find the equation of the function.

PRACTICE EXERCISES

3. WRITING MATH In the example on page 550, will youobtain the same equation if you select three differentpoints? Explain.

4. PHYSICS The relationship between two variables in anexperiment can be represented by the parabola shown.Using three points from the graph, find the equation ofthe parabola.

5. A parabola contains the points (1, �4), (0, �3), (2, �3),(3, 0) and (�2, 5). Choose three points and find theequation of the parabola.

Find the equations for each parabola.

6. 7.

8. CHAPTER INVESTIGATON Calculate your vehicle’s average rate of speed infeet per second for each run of the ramp. Make a poster showing a diagramof the vehicle and the vehicle’s speed for various inclines of the ramp. Is therea relationship between the increase in the incline and the vehicle’s speed?


Factor the following trinomials. (Lesson 11-9)

9. 15a 2 � 14a � 8 10. a 2 � 5a � 14 11. a 2 � 11a � 24

12. 12a 2 � 5a � 2 13. 3a 2 � 14a � 24 14. 4a 2 � 16a � 16

y

x

12108642

�2�4�6�8

�10�12

4 8 �8 �2

(3, �9)

y

x

4

2

�2

�4

�6

�8

1 3 �3 �1

Lesson 12-7 Problem Solving Skills: Graphs to Equations 551

y

x

4

�2

�4

�6

�8

4 �4 �2

Five-stepPlan

1 Read2 Plan3 Solve4 Answer5 Check

y � x2 � 2x � 5

y � 2x2 � 1

y � �2x2 � 2x � 3

y � x2 � 2x � 3

y � �x2 � 6x � 8

Answers will vary.

(5a � 2)(3a � 4)

(4a � 1)(3a � 2)

(a � 2)(a � 7)

(a � 6)(3a � 4)(a � 8)(a � 3)

4(a � 2)(a � 2)

y � �x2 � 4x � 12

3. Yes. Any threenon-collinear pointslocated on the graphwill work. However,you may find it easierto solve the systemof equations if youuse the y-interceptas one of the points.


Chapter 12 ReviewVOCABULARY

Match the letter of the word in the list at the right with the description on the left.

1. an equation that contains an x2 term and involves no term with a higher power of x

2. the relationship in which each x-value has a unique y-value

3. the lowest point on a parabola that opens upward or the highest point on a parabola that opens downward

4. a function of the form f(x) � ax2 � bx � c, where a � 0

5. the line that divides a parabola in half

6. the graph of a quadratic function

7. used to find the length of a segment given the coordinates of the endpoints

8. solving a quadratic function by adding ��b2��2

to both sides of therelated equation

9. express a polynomial as a product of monomials, binomials, or polynomials

10. formula used to solve quadratic functions in the form ax2 � bx � c � 0

LESSONS 12-1 and 12-2 Quadratic Functions, p. 520

� To graph a quadratic function, find at least five ordered pairs by selecting x-values and finding the corresponding y-values. Locate and draw a smoothcurve through the points.

� To locate the vertex, use x � ��2ba� to find the x-coordinate. Then substitute the

x-value into the equation to find the y-coordinate.

� The axis of symmetry for the graph passes through the vertex and is parallel tothe y-axis.

Graph each function. Give the coordinates of the vertex for each graph.

11. y � �3x 2 12. y � 4x 2 � 4x 13. y � �2x 2 � 4x � 1

Each graph below is in the form y � ax2 � c. Write an equation for each parabola.

14. 15. 16.

a. axis of symmetry

b. complete thesquare

c. distance formula

d. factoring

e. function

f. general quadraticfunction

g. midpoint formula

h. parabola

i. quadratic equation

j. quadratic formula

k. vertex

l. x-intercept

�2

�4

�6

2

�2�4 2 4

y

x

�2

�4

2

4

�2�4 2 4

y

x2

4

6

�2�4 2 4

y

x

�

Chapter 12 Review 553

� A function of x assigns each x-value exactly one y-value. If each vertical line inthe coordinate plane contains at most one point of the graph, the graphrepresents a function.

Determine if each graph is the graph of a function.

17. 18. 19.

LESSON 12-3 Factor and Graph, p. 530

� The x-intercepts are the solutions of the related quadratic equation. Thenumber of x-intercepts and approximate solutions may be found by graphing.To find the exact solutions, let y � 0, and factor the equation.

Use a graphing calculator to determine the number of solutions for eachequation. Factor to solve each equation.

20. x 2 � 14x � �49 21. x 2 � 63 � �2x 22. 3x 2 � 9x � 84 � 0

The length of a Rugby League field is 52 m longer than its width w.

23. Write an expression of the area of the field.

24. The area of a Rugby League field is 8160 m2. What are the dimensions of the field?

LESSON 12-4 Complete the Square, p. 534

� To complete the square for an expression in the form ax 2 � bx, add ��b2

��2.

� To solve an equation by completing the square, rewrite the equation in the

form ax 2 � bx � c, add ��b2

��2

to each side of the equation, factor, and simplify.

Solve each equation by completing the square. Round to the nearest tenth if necessary.

25. x 2 � 2x � 8 � 0 26. x 2 � 8x � �15 27. x 2 � 14x � �49

28. x 2 � 4x � 12 � 0 29. d 2 � 3d � 10 � 0 30. y 2 � 19y �4 � 70

31. d 2 � 20d � 11 � 200 32. a 2 �5a � �4 33. p 2 � 4p � 21

34. x 2 � 4x � 3 � 0 35. d 2 � 8d � 7 � 0 36. r 2 � 10r � 23

LESSON 12-5 The Quadratic Formula, p. 540

� The quadratic formula can be used to solve any quadratic equation of theform ax 2 � bx � c � 0, a � 0.

The quadratic formula is x � .�b � �b2 � 4�ac��

2a

y

x

y

x

y

x



37. 2x 2 � 5x � 3 38. 6x 2 � x � 1 39. x 2 � 2x � 6

40. x 2 � 3x � 18 � 0 41. v 2 � 12v � 20 � 0 42. 3t 2 � 7t – 20 � 0

43. 5y 2 � y � 4 � 0 44. x 2 � 25 � 0 45. 2x 2 � 98 � 28x

46. 4r 2 � 100 � 40r 47. 2t 2 � t � 14 � 0 48. 2n 2 � 7n � 3 � 0

LESSON 12-6 The Distance Formula, p. 544

� The distance, d, between any two points (x1, y1) and (x2, y2) on the coordinateplane may be found using the distance formula:

d � �(x2 ��x1)2 �� (y2 �� y1)2�

The coordinates of the midpoint of a line segment with endpoints

(x1, y1) and (x2, y2) are ��x1 �

2x2

�, �y1 �

2y2

��.

Find the midpoint of the segment with the given endpoints. Then, find thedistance between each pair of points to the nearest tenth.

49. L(�10, 5), M(4, �9) 50. C(6, 4), D(�8, �6) 51. P(0, �7), Q(12, 2)


52. (12, 3), (�8, 3) 53. (0, 0), (5, 12) 54. (6, 8), (3, 4)

55. (�4, 2), (4, 17) 56. (�3, 8), (5, 4) 57. (9, �2), (3, �6)

58. (�8, �4), (�3, �8) 59. (2, 7), (10, �4) 60. (4, 2), �6, ��23��61. �5, �14��, (3, 4) 62. ��45�, �1�, �2, ��12�� 63. �3, �37��, �4, ��27��

LESSON 12-7 Problem Solving Skills: Graphs to Equations, p. 550

� Work backwards to write the equation of a parabola. Use the general equationy � ax 2 � bx � c and three points on the graph.

64. A parabola contains the points (�4, 23), (0, �1) and (2, �49). Find anequation for the parabola.

Find the equation for each parabola.

65. 66.

CHAPTER INVESTIGATION

EXTENSION Write a report about your experiment. Describe your vehicle andexplain why you built it the way you did. Explain what happened when you used the ramp and changed its angle. Include all of the data you collected in your report.

2

�2

�4

�6

�8

�1�3�5 1 3 5

y

x

4

�4

8

12

�2�6�10 2 6 10

y

x

Chapter 12 AssessmentGraph each function for the domain of real numbers. Give the coordinates ofthe vertex for each graph.

1. y � �2x 2 2. y � 3x 2 � 4 3. y � 7 � �3x 2 � 2x

Determine if each graph is a function.

4. 5. 6.

Factor to solve each equation.

7. x 2 � 6x � 16 � 0 8. x 2 � 9 � 6x 9. 4x 2 � 4x � � 1


10. x 2 � 14x 11. x 2 � 8x 12. x 2 � x

Solve by completing the square.

13. x 2 � 3x � 2 � 0 14. x 2 � 4x � 12 15. 2x 2 � 2x � 40


16. 3x 2 � 4x � 15 17. x 2 � 8x � �3 18. 24x � �x 2 � 136

Calculate the distance between each pair of points to the nearest tenthof a unit.

19. A(1, �5), B(�3, 7) 20. L(�6, 4), M(2, �2) 21. R(0, 8), S(�3, �12)

Find the midpoint of the segment with the given endpoints.

22. G(�13, 9), H(7, �11) 23. P(10, �6), Q(�4, �7) 24. C(�5, 3), D(12, 8)

Solve.

25. How long will it take a ball thrown upwards with an initial velocity of 72 ft/sec to reach the ground?

Find the equation for each parabola.

26. 27.

2

4

6

8

�2�4 2 4

y

x

2

4

�2

�4

�6

�2�4 2 4

y

x

y

x

y

x

y

x

Chapter 12 Assessment 555mathmatters3.com/chapter_assessment

http://mathmatters3.com/chapter_assessment


Standardized Test Practice5. Solve the proportion: �x

2�x

1� � �89�. (Lesson 7-1)

�45� �

78�

�11

01� �

97�

6. Determine whether the two triangles below are similar. If so, by which postulate? (Lesson 7-4)

Angle Angle Postulate

Side Angle Side Postulate

Side Side Side Postulate

The triangles are not similar.

7. Triangle RST is to be reflected over the y-axis. What will be the coordinates of S�? (Lesson 8-1)

(�3, �1) (3, 1)

(3, �1) (�3, 1)

8. What is the value of z? (Lesson 8-5)

� � � � � � � ��12 0

4 5DC

BA

xz

wy

�31

3�2

24

10

DC

BA

2 4�2�4 x

y4

2

�2

�4

R

S T

D

C

B

A

12 16

20

18 24

30

DC

BA

Test-Taking TipQuestion 8In Question 8, you are only asked to solve for z. It is notnecessary to find the product of the two matrices.

Part 1 Multiple Choice

Record your answers on the answer sheetprovided by your teacher or on a sheet of paper.

1. Refer to the diagram below. Which pair ofangles is supplementary? Assume thetransversal is not perpendicular to the parallellines. (Prerequisite Skill)

�7 and �4 �8 and �6

�1 and �6 �7 and �2

2. The figure below is an isosceles trapezoid withA�B� � D�C�. What is m�C? (Lesson 4-9)

22° 68°

86° 112°

3. A 16-oz jar of spaghetti sauce costs $3.59. If the cost per ounce is the same, how muchwould a 40-oz jar cost? (Lesson 5-1)

$7.95 $8.98

$9.25 $10.77

4. What is the solution to this system ofequations? (Lesson 6-4)

3x � y � �112x � 2y � �10

(4, �23) (�6, 7)

�1�12�, 3� (�3, �2)DC

BA

DC

BA

DC

BA

A112°

B

CD

DC

BA

1 2

3 48 5

7 6

Chapter 12 Standardized Test Practice 557mathmatters3.com/standardized_test

Preparing for Standardized TestsFor test-taking strategies and morepractice, see pages 709–724.

Part 2 Short Response/Grid In

Record your answers on the answer sheetprovided by your teacher or on a sheet of paper.

9. Mr. Russell purchased an antique tin for$24.50 at a flea market. After cleaning thepiece, he sold it to a collector for $800. Whatis the percent increase of the cost of the tin?Round to the nearest whole percent.(Prerequisite skill)

10. Find the value of xfor the quadrilateral.(Lesson 4-7)

11. If ABCD is a trapezoid with median E�F�, what is the length of E�F�? (Lesson 4-9)

12. Find the volume in cubic inches of the three-dimensional figure. (Lesson 5-7)

13. Jill and Gena went shopping and spent $122 altogether. Jill spent $25 less than twice as much as Gena. How much did Gena spend? (Lesson 6-4)

14. Find x. Round to the nearest tenth ifnecessary. (Lesson 7-5)

15. A card is drawn at random from a standarddeck of cards. What is the probability that thecard drawn is red or a 10? (Lesson 9-1)

4

4 7.5

7.5

18

x

30 in.

18 in.

6 in. 12 in.

8 in.

D C

E F

A B

16 m

28 m

77�

98�

80�

x�

16. Write an expression for the area of the figure. (Lesson 11-3)

17. The area of a rectangular playground is 750 m2.The length of the playground is 5 m greaterthan its width. What are the length and widthof the playground? (Lesson 12-5)

18. The function h(t) � �16t2 � v0t � h0

describes the height in ft above the ground h(t) of an object thrown verticallyfrom a height of h0 ft, with an initial velocityof v0 ft/sec, if there is no air friction and t isthe time in seconds that it takes for the ball to reach the ground. A ball is thrown upward from a 100-ft tower at an initialvelocity of 60 ft/sec. How many seconds will it take for the ball to reach the ground?(Lesson 12-5)

Part 3 Extended Response

Record your answers on a sheet of paper.Show your work.

19. Paris’s family is building a house on a lot that is 91 ft long and 158 ft wide.(Lesson 2-6)

a. The town law states that the sides of ahouse cannot be closer than 10 ft from the edges of a lot. Write an inequality to represent the possiblelengths of the house.

b. They want their house to be at least 2800 ft2 and no more than 3200 ft2. Theyalso want the house to have the maximumpossible length. Write an inequality for the possible widths of the house. Round to the nearest whole number of feet.

6x � 4

4x � 12x

x

http://mathmatters3.com/standardized_test

chapter 12: quadratic functions

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