Memorize the following!

Three forms of a quadratic equation.Standard: y = ax + bx + cVertex: y = a(x – h) + k

Intercept: y = a(x – p)(x – q)X coordinate of vertex:

-b2a

2

2

VERTEX FORM

+ k

If |a | > 1, parabola narrowsIf |a | < 1, parabola widensIf a < 0, parabola opens down

THE PARENT FUNCTIONof the quadratic family

𝑦=𝑥2Characteristics:

shape is parabolic and symmetricaxis of symmetry is verticalhas a maximum or a minimum value

Vertex is (h, k)

Graphing from vertex form

1. Plot the vertex ( h, k )2. Determine if the parabola opens up or down3. Draw the axis of symmetry as a dashed vertical line4. Choose an x value on one side of the vertex, 5. Substitute this value in, solve for y and plot the point.6. Plot the reflection image of the point.7. Repeat steps 5 and 6 for an additional point8. Sketch the parabola.

STANDARD FORM

Effect of a is the same. If a is positive the parabola opens up, if negative, down.Axis of symmetry is the vertical line x = -b

2a

Vertex is (,f(x))

To complete graph, choose an x value near the vertex, calculate its corresponding y value, plot the point, then reflect it over the line of symmetry.

+ bx + c

The y-intercept is (0, c). Plot this point, then reflect it over the line of symmetry.

- or-

Graphing from standard form• Plot the vertex, the x coordinate is found using Find the y coordinate by plugging the x value into the original function and solving for y. -Determine if the parabola opens up or down-Sketch the axis of symmetry using a dashed line.-Choose a value for x, plug it into the equation and solve for y. Plot this point-Plot the reflection image of this point-Choose another x value and repeat the last 2 steps-Sketch the parabola

ab2

INTERCEPT FORM

Effect of a is the same.

The x-intercepts are p and q.

The axis of symmetry goes through the x-axis at the midpoint of the segment defined by p and q.

The vertex is (x, f(x)), where

𝑦=𝑎(𝑥−𝑝)(𝑥−𝑞)

Graphing from intercept form

1. Plot the x intercepts p and q2. Draw the axis of symmetry halfway

between p and q3. The x coordinate of the vertex is

the x value of the axis of symmetry, plug this into the equation and solve for y. Plot the vertex.

4. Sketch the graph.

The path of a baseball after it is hit is modeled by the function

30032. 2 ddh

h is the height of the baseball in feet and d is the distance in feet the baseball is from home plate. What is the maximum height reached by the baseball? How far is the baseball from home plate when it reaches it’s maximum height?

QUADRATIC FUNCTIONSThe Golden Gate Bridge in San Francisco has two towers that rise 500 feet above the road and are connected by suspension cable as shown. Each cable forms a parabola with the equation

+ 82)2100(89601

xy

What is the distance between the two towers if the cable supports are 500 feet above the road?

What is the height above the road of a cable at its lowest point?

• Although a football field appears to be flat, its surface is actually shaped like a parabola so that rain runs off to either side. The cross section of a synthetic field can be modeled by the equation

+ 1.5 2)80(000234.0 xy

Where x and y are measured in feet.

What is the field’s width?

What is the maximum height of the field’s surface?

A kernel of popcorn contains water that expands when the kernel is heated, causing it to pop. The equations below give the “popping volume”, y (in cubic cm/gram) of popcorn with the moisture content as x (as percent of the popcorn’s weight).

Hot-oil popping:

Hot-air popping: – 94.8

1. For hot-air popping, what moisture content maximizes popping volume?

2. What is the maximum volume?

3. For hot-oil popping, what moisture content maximizes popping volume?

4. What is the maximum volume?

5. The moisture content of popcorn typically ranges from 8% to 18%. Examine the graph of both equations for the interval

6. Based on a comparison of both graphs, what general statement can you make about the volume of popcorn produced from hot-air popping versus hot-oil popping for any moisture content in the interval

Writing Quadratic Functions in Standard Form

Multiply using FOILcombine like termsdistribute the -1283

)283(

)2847(

)7)(4(

2

2

2

xx

xx

xxx

xxy

1382

5882

5)44(2

5)422(2

5)2)(2(25)2(2

2

2

2

2

2

xx

xx

xx

xxx

xxxy

You have 400 feet of fencing with which you need to enclose a rectangular field along a straight river bank. (There is no need to fence along the river.)

Let x represent the width of the field and write an equation to represent the area of the field.

Find the dimensions of the field with the maximum area.

Warm-up

Anne and Linda went out to lunch. When the bill came Anne realized she had only 8 dollars in her wallet. Linda told her not to worry because she had enough money to cover three fifths of the bill. With Anne’s eight dollars and Linda’s money they could pay. How much was the lunch bill? How much did Linda contribute?

Factor the following expressions.

x2 – x - 6 x2 -3x

Factoring

- Factor out the GCF

- Look for special patterns such as the difference of two squares, perfect square

trinomial- FOIL to check your answers!

Undo FOIL!

3103

32

149

107

65

2

2

2

2

2

xxy

xxy

xxy

xxy

xxy

Factor out any GCF first!

xxy

xxy

wy

2

2

2

10082

305

Difference of Squares

494

12125

322

9

2

2

2

2

xy

xy

xy

xy

Perfect Square Trinomials

4129

96

11449

2

2

2

xxy

xxy

rry

Solving quadratic equations by factoring

- Set the equation equal to zero .(You’re making y zero because you’re finding the x intercepts.)

-Factor the equation completely.

-Set each factored portion equal to zero and solve for the variable.

Zero Product PropertyIf A and B are real numbers or algebraic expressions, If AB = 0 then A = 0 or B = 0

In Quadratic equations the x intercepts are also called

-solutions-roots

-zeros of the function

When solving, instructions might say:-Solve the equation.

-Find the function’s zeros.-Write the quadratic function in intercept form

and find the function’s zeros.

You are making a coffee table with a glass top surrounded by a cherry border. The glass is 3 ft. by 3 ft.. You want the cherry borer to be a uniform width. You have 7 square feet of cherry. What should the width of the border be?

The dimensions of a rectangular garden were 5 meters by 12 meters. Each dimension was increased by the same amount. The garden then had an area of 120 square meters. Find the dimensions of the new garden.

A graphic artist is designing a poster that consists of a rectangular print with a uniform border. The print is to be twice as tall as it is wide. The border is to be 3 inches wide. If the area of the poster is to be 680 square inches find the dimensions of the print.

• You have made a rectangular stained glass window that is 2 feet by 4 feet. You have 7 square feet of clear glass to create a border of uniform width around the window. What should the width of the border be?

Warm-up Solve the following equations by factoring.

16x2 = 8x -1

y = 25x2 -4

15 = 3x2 -12x

5x2 = 30x

Factor the following expressions.

X2 + 2x + 1

X2 – 4x + 4

4x2 +12x + 9

Solve the following equations.

X2 -10x + 25 = 0 x2 – 16x = -64

warm-up

If x2 = 16 what is the value of x?

Product Property Quotient Property

A square-root expression is simplified if:-No radicand has a perfect square factor other than 1-There is no radicand in the denominatorSimplify

√12 √45 √6 . 2√6 √98

Rationalizing the denominator-the process of eliminating a radical from the denominator.

baab ba

ba

Solving quadratic equations by taking square roots.

• Rewrite the equation so the squared term is isolated on one side of the equal sign.

• Take the square root of each side.• Remember there will be a positive and a

negative root.• Ex.

22

8

8

162

1712

2

2

2

2

x

x

x

x

x

You can use square roots to solve some quadratic equations.

-you can use this method when there is no linear term

Ex. 2x2 + 1 = 17 1/3(x + 5)2 = 7

Remember + and - !!!!!!

The height h(in feet) of the object t seconds after it is dropped can be modeled by the function:

h = -16t2 + ho

(ho is the object’s initial height)A stunt man working on the set of a movie is to fall out of a

window 100 feet above the ground. For the stunt man’s safety, an air cushion 26 feet wide by 30 feet long by 9 feet

high is positioned on the ground below the window.For how many seconds will the stunt man fall before he

reaches the cushion?A movie camera operating at a speed of 24 frames per

second records the stunt man’s fall. How many frames of film show the stunt man falling?

From 1990 to 1993 the number of truck registrations (in millions) in the United States can be approximated by the model R = .29t2 + 45 where t is the number of years since 1990. During which year were approximately 46.16 million trucks registered?

In 1992 the average income I(in dollars) for a doctor aged x years could be modeled by:

I = -425x2 + 42,500x – 761,000For what ages did the average income for a doctor exceed $250,000?

The aspect ratio of a TV screen is the ratio of the screen’s width to its height. For most TVs the aspect ratio is 4:3. What are the width and height of the screen for a 27 inch TV? Use the Pythagorean Theorem to solve.

SOLVE USING

2 𝑥2+1=17 = 21

2 𝑥2=16𝑥2=8

𝑥=±2√2

=

=

𝑥+5=±3√7 =

𝑥=−5±3 √7

QUADRATICSTHERE’S AN APP FOR THAT!

The tallest building in the United States is in Chicago, Illinois. It is 1450 tall.a) How long would it take a penny to drop from the top of this building?

b) How fast would the penny be travelling when it hits the ground if the speed is given by s = 32t where t is the number of seconds since the penny was dropped?

When an object is dropped, its speed continually increases, and therefore its height above the ground decreases at a faster and faster rate. The height, h (in feet) of the object t seconds after it is dropped can be modeled by the function

+ Where is the object’s initial height.

WHAT IF THE REAL NUMBER SYSTEM ISN’T SUFFICIENT?

𝑥2=−9 𝑥2+7=4 -

WE INVENT SOME MORE NUMBERS!

IMAGINARY NUMBERS

i =

i is the imaginary unit it is equal to the square root of -1

i = = 6i =i

A complex number in standard form - a + bi

Ex. 2 + 3i

A pure imaginary number 2i

1 36 5 5

Adding and Subtracting Complex Numbers

• The imaginary number i acts like a variable but when you get substitute -12i

iii

7)23()4(

iiiii

22432

)24()32(

Multiplying Complex Numbers

iiii

iii

6896196

)3)(3()3(

2

2

Substitute -1 for i squared

Complex conjugates example (5 + 3i) and (5 – 3i)

The product of complex conjugates is always a real number.

Dividing Complex Numbers

i

iiiiiii

ii

ii

ii

513

51

5131422163105

2121

2135

2135

2

2

QUIZ 5.4

1. Simplify:

Perform the indicated operation: 2. (25 + 15i) + (25 – 6i) 3. (25 + 15i) - (25 – 5i)4. (5 + i)(8 + i)5.

6. Write the complex conjugate of 6 – 3i.7. Solve: = 120

YET ANOTHER WAY TO SOLVEQUADRATIC EQUATIONSCompleting the square

-Use inverse operations to move the constant term to the other side of the equal sign-Divide the coefficient of the linear term in half and square it, add it to both sides of the equal sign-Factor the now perfect square trinomial-Solve by taking the square root of both sides

Complete the expression to make it a square:

𝑥2+6 𝑥+¿¿ - 22x + ___ + x + ___

Rewrite each expression as the square of a binomial

(𝑥+3)2 (𝑥−11)2 (𝑥+425

)2

COMPLETE THE SQUARETO SOLVE THE EQUATION

+ 6x – 8 = 0Move -8 to other side + 6 x = 8

Add half of the middle term squared to both sides of the quadratic equation

+ 6x + 9 = 8 + 9

Simplify + 6x + 9 = 17

Factor to rewrite trinomial square as square of binomial

= 17

both sides x + 3 =

Isolate x 𝑥=−3 ±√17

COMPLETING THE SQUARE (cont’d)The leading coefficient must be one. Divide by the coefficient of if necessary.

Example:

𝑥2− 65 𝑥=¿85

𝑥2− 65 𝑥+925=

85 +

925

=

𝑥− 35=±√ 4925𝑥=

35 ±75

𝑥=2 ,𝑥=− 45

5 𝑥2−6 𝑥=8

WRITING A QUADRATIC EQUATIONIN VERTEX FORM

+ k

Example:

Make room for c 𝑦=𝑥2+6 𝑥+16Complete the square 𝑦+9=𝑥2+6 𝑥+9+16Rewrite trinomial + 16

Isolate y + 16 -9

Simplify + 7

Write in vertex form + 7To name the vertex:

Vertex is (-3, 7)

On dry asphalt the distance (d) in feet needed for a car to stop is given by

d = .05s2 + 1.1sWhere s is the car’s speed in miles per hour. What speed limit should be posted on a road where drivers round a corner and have 80 feet to stop?

You want to plant a rectangular garden along part of a 40 foot side of your house. To keep out animals you will enclose the garden with wire mesh along its three open sides. You will also cover the garden with mulch. If you have 50 feet of mesh and enough mulch to cover 100 square feet, what should the garden’s dimensions be? Solve by writing a quadratic equation and completing the square.

house

garden

40 ft.

x

x

50-2x

THE QUADRATIC FORMULAThe “catch-all” method for solving quadratic equations

1. Start with the standard form of quadratic equation:

2. Determine values of a, b and c.

3. Replace a, b and c in quadratic formula with values from equation.

4. Solve for x.

a= -7b = 2c= 9

aacbbx

242

)7(2)9)(7(422 2

x

𝑥=−2±√4−(−252)

−14

𝑥=−2±√256−14

𝑥=−2±16−14

x = -1, x = 9/7

-7x² + 2x + 9 = 0

EXAMPLESSOLVE USING THE QUADRATIC FORMULA

2. x²+ 2x + 1 = 0

3. x² − 6x + 10 = 0

1. 3x² - 5x = 2

X = 2 or

X = -1

X = 3

THE DISCRIMINANT(“the qualifier” of solutions)

b² − 4acSTANDARDEQUATION DISCRIMINANT #/TYPE OF SOLUTIONS SOLUTION(S)

x² − 6x + 10 = 0

2/Real

1/Real

-1 and 9/7

-1

2/Imaginary 3 ± ί(-6)²−4(1)(10)

-7x² + 2x + 9 = 0 2²−4(-7)(9)

+/0/−

+

x²+ 2x + 1 = 0 2²− 4(1)(1) 0

−

=256

=0

=-4

If the discriminant is positive there are 2 real solutions.If the discriminant is negative there are 2 imaginary solutions.If the discriminant is zero there is one real solution.

HOMEWORK QUIZQUADRATIC FORMULA

Solve each equation using the quadratic formula.

1. -2, 7

State the discriminant, then indicate the number and type of solution(s):

-24; 2 Imaginary 97; 2 Real 0; 1 Real

Objects thrown or launched00

216 htvth

A baton twirler tosses a baton in the air. It leaves the twirler's hand 6 feet above the ground and has an initial vertical velocity of 45 feet per second. The twirler catches the baton when it falls back to a height of 5 feet. For how long is the baton in the air?

ending initial beginning height vertical height

velocity

• An astronaut standing on the moon throws a rock into the air with an initial vertical velocity of 27 ft/sec. The astronaut’s hand is 6 feet above the moon’s surface. The height of the rock is given by

• How many seconds is the rock in the air?• How many seconds would the rock be in the

air when thrown from the earth’s surface?

6277.2 2 tth

• A basketball player passes the ball to a teammate who catches it 11 feet above the court, just above the rim of the basket, and slam-dunks it through the hoop. The first player releases the ball 5 feet above the court with an initial vertical velocity of 21 ft./sec. How long is the ball in the air before being caught, assuming it is caught as it rises?

Use the quadratic formula to solve!

• For the years 1989-1996 the amount A(in billions of dollars) spent on long distance telephone calls in the United States can be modeled by

where t is the number of years since 1989. In what year did the amount spent reach $60 billion?

51488.56. 2 ttA

• Niagara Falls in New York is167 feet high. How long does it take for water to fall from the top to the bottom of Niagara Falls?

ohth 216

• A rectangle has an area of 60 square feet. The length is one foot more than the width. Use the quadratic formula to determine the dimensions of the rectangle to the nearest hundredth.

• A child tosses a ball upward with a starting velocity of 10 ft/sec from a height of 3 feet. If the ball is not caught, how long will it be in the air? Round your answer to the nearest tenth.

QUADRATIC INEQUALITIES

1. Graph the boundary - broken line parabola

2. Determine where the solutions to the inequality are in relation to the boundary by testing a point.

y > x2 + 2x – 8

3. Shade the appropriate area

EXAMPLE #2

y < 2x2 – 3x + 1

1. Graph the boundary - solid line parabola

2. Determine where the solutions to the inequality are in relation to the boundary.

3. Shade the appropriate area.

SYSTEM OF QUADRATIC INEQUALITIES

1. Graph each inequality.

2. If the graphs overlap, the solution set is the intersection of the graphs.

Example:

y < – x2 + 4

y > x2 – 2x – 3

YOUR TURN

Graph the system of inequalities consisting of y ≥ x2 and y < 2x2 + 5.

Solving quadratic inequalities algebraicallyExample 1 (one variable inequality):

Change the inequality to = and solve:

Prepare the number line:

x < -3 or x > 4

This problem could also be solved by examining the corresponding graph of . Graph the quadratic (parabola) by hand or with the use of a

graphing calculator.The quadratic is

greater than zero where the graph is ABOVE

the x-axis

For > identify the x values for which thegraph lies above the x axis.

For < identify the x values for which the graphlies below the x axis.

Try these!

499

47

082

34

2

2

2

2

x

xx

xx

xx

To solve a quadratic inequality algebraically:-find the zeros by solving the equation set = to zero

-plot the zeros on a number line

-pick a value and test it to see if the solutions lie inside the 2 values or outside of the 2 values

-mark the solutions on the number line as a segment between two values or two rays going in opposite

directions, outside of two values

-state your answer as an inequality

Writing a quadratic function in vertex form from a graph

• 1. Plug the vertex into + k• 2. Choose another point on the parabola and

plug it into the equation for x and y so you can solve for a.

• 3. Plug the value of a into the equation from step 1.

Try this!

Writing a quadratic function in intercept form from a graph.

• 1. Read the x intercepts off of the graph and plug them into y = a(x – p)(x – q)

• 2. Choose another point off of the graph and plug the x and y coordinates into the equation and solve for a.

• 3. Substitute a into the equation from step 1.

Try this!

Warm up

• Solve using the method of your choice.

5)2(30

02610

1325

058

2

2

2

2

x

xx

x

xx

Finding a quadratic model for a data set

• -Use the stat key to enter the data into lists one and two

• STAT- CALC- #5QUADREG- enter – enter• Plug the a, b and c into + bx + c• to graph y= - Vars - #5 Statistics – enter – EQ –

enter - graph

Find a quadratic model to fit the following data

Diameter (mm)

20 25 30 35 40 45 50

Time (min)

27 42 61 83 109 138 170

The time it takes to boil a potato based on the diameter