�1

U2 - Parent Graphs Name ________________________

- In mathematics, you see certain graphs over and over again. For that reason, these original, common functions are called _______________ _______________ , and they include graphs of quadratic functions, square roots, absolute values, cubics, and cube roots.

- When you translate, reflect, or otherwise change the original graph, you are _______________ the parent graph.

Odd/Even

- Most of the parent graphs have special characteristics, such as symmetry. If a parent graph is symmetrical about the _______________ , it is said to be even. If the parent graph is symmetrical about the _______________ , it is said to be odd.

- Notes from video:

Determine whether the function is even, odd, or neither.

a. � b. � c. �g x( ) = x3 − x h x( ) = x2 +1 f x( ) = 5 − 3x

�2d. � e. � f. �

g. � h. �

Graphing Form

- Every parent graph has its own _______________ form. This form is ideal for going straight to the graph or finding the key points of the graph without graphing. The basic set up for the graphing form is: � , where a, h, and k are all parameters that affect the graph in different ways.

- a tells you if the graph is vertically ____________________ or ____________________ . Also, if a is positive the graph opens ____________________ and if its negative the graph opens ____________________ .

- h tells you if the graph is shifted to the ____________________ or ____________________ .

- k tells you if the graph is shifted ____________________ or ____________________ .

Graphing Forma. � _________________________

b. � _________________________

c. � _________________________

d. � _________________________

e. � _________________________

f. � _________________________

g x( ) = x4 − x2 −1 h x( ) = 2x3 + 3x f x( ) = x6 − 2x2 + 3

g x( ) = x3 − 5x f x( ) = x 1− x2

y = a(x − h)n + k

y = x2

y = x3

y = 1x

y = x

y = x

y = bx

�3- When the equation of the parabola is not in graphing form there is a short cut to getting it in graphing form

quickly, it’s called ___________________________________ .

Complete the Square of the following.

1. � 2. � 3. �

4. � 5. � 6. �

7. � 8. � 9. �

10. � 11. � 12. �

13. � 14. � 15. �

16. � 17. � 18. �

y = x2 − 2x −15 y = x2 + 8x +10 y = x2 + 2x + 4

y = x2 + 4x + 9 y = x2 + 5x + 2 f x( ) = x2 − 4x +11

f x( ) = x2 + 5x + 2 f x( ) = x2 − 7x + 2 f x( ) = x2 +10x − 3

y = x2 − 6x +11 y = x2 − 2x − 9 y = x2 +16x +14

y = x2 + 26x + 68 y = x2 − 3x − 2 y = x2 + 7x −1

y = −x2 + 20x − 80 y = −x2 −14x − 47 y = 3x2 −12x +1

�4

19. � 20. � 21. �

Parent graphs w/ transformations

- Many functions have graphs that are transformations of the _______________ _______________ .

If you are working with the parent graph � , you could move the graph up two units by _______________ .

Similarly, you can move the same parent graph right two unit by _______________ .

Here are the rules for vertical and horizontal shifts:

Use the graph of � to sketch a graph of each function.

a. � b. �

y = −2x2 − 2x − 7 y = 1.4x2 + 5.6x + 3 y = 23x2 − 4

5x

f x( ) = x2

f x( ) = x3

g x( ) = x3 −1 h x( ) = x + 2( )3 +1

-2

-1

0

1

2

� x � f x( )

�5- Another type of transformation is a _______________ .

- For instance, if you let the x-axis be a mirror, the the graph � is the mirror image (or reflection) of the parent graph of �

Here are the rules for reflections:

- In the form, � , if a is _______________ the graph will flip over the _______________ .

The following is a graph of the function � .

These graphs are transformations of the parent graph. Write equations for each of therm.

a. b. c.

_______________________ _______________________ _______________________

- Remember when you deal with graphs of square roots you must restrict the domain to exclude __________________________ numbers inside the radical.

Nonrigid Transformations

- Horizontal shifts, vertical shifts, and reflections are _______________ _______________ because the shape of the graph is _______________ .

- These transformations change only the _______________ of the graph in the coordinate plane.- _______________ _______________ are those that cause a _______________ —a change in the shape

of the original graph.

Here are the rules for nonrigid transformations:- In the form, � , if � the graph is _______________ , and if � the graph is

_______________ .

h x( ) = −x2

f x( ) = x2

y = a x − h( )n + k

f x( ) = x4

y = a x − h( )n + k a >1 a <1

�6Compare the graph of each function with the graph � .

a. �

b. �

- In this unit, we deal specifically with parabolas, which has a basic graphing form of, ______________ . So, lets go over how the parameters a, h, and k specifically affect parabolas.

- a � _______________ _______________ - If a is negative the graph opens _______________ , and if a is positive the graph opens

_______________ . - If � the graph is _______________ , and if � the graph is _______________ . - h � _______________ _______________- k � _______________ _______________- _______________ is the center of the parabola, the lowest or highest point depending on which way the

graph opens, which can be easily found if equation is already in graphing form. It is (-h, k).

Write an equation for each parabola described below.

1. A parabola that opens down and is shifted up 12 units from the origin.

2. Take the previous parabola and make it open up and compress it by .02.

3. A parabola that is compressed by � with a vertex at (3, 4).

4. Take the previous parabola and make it open down and shift it 7 units to the left and 8 units down.

5. A parabola that opens down and its vertex is (-5, 2).

f x( ) = x

h x( ) = 3 x

g x( ) = 13x

→

a >1 a <1→→

23

�76. A parabola that is shifted 15 units to the right and 35 units up from the origin.

7. Take the previous parabola and make it open down and compressed it by 8.

8. Take the previous parabola and shift it 20 units to the left and 42 units down.

9. A parabola that opens up and has a vertex of (1, 2).

10. Take the previous parabola and make it open down and stretch it by 9.

11. A parabola that opens up, has a vertex at (2,0), and is stretched by 3.

12. Take the previous parabola and make it open down and compress it by .5.

13. A parabola that opens down is shifted 4 units to the left and 8 units up from the origin.

14. Take the previous parabola and make it open up and stretch it by 5.

15. Take the previous parabola and compress it by � and shift it 5 units right and 4 units up.

16. A parabola that is stretched by 7 and has a vertex of (-2, 11).

17. Take the previous parabola and compress it by 3 and make it open down.

18. Take the previous parabola and shift it 7 units to the right and 6 units down.

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�8Parabola word problems (rabbits, etc.)

1. Jumping Jackrabbits- The diagram at right shows a jackrabbit jumping over a three-foot-high

fence.  To just clear the fence, the rabbit must start its jump at a point four feet from the fence. 

- Sketch the path of the jackrabbit.  Choose where to place the x- and y‑axes in your diagram so that they make sense and make the problem easier.  Label as many points as you can on your sketch.

- Find the missing parameter and write the equation that fits the path of the jackrabbit.

2. When Ms. Bibbi kicked a soccer ball, it traveled a horizontal distance of 150 feet and reached a height of 100 feet at its highest point.  Sketch the path of the soccer ball and find an equation of the parabola that models it.

3. At the skateboard park, the hot new attraction is the U-Dip, a cement structure embedded into the ground.  The cross-sectional view of the U-Dip is a parabola that dips 15 feet below the ground.  The width at ground level, its widest part, is 40 feet across.  Sketch the cross-sectional view of the U-Dip, and find an equation of the parabola that models it.

�94. A fireboat in the harbor is helping put out a fire in a warehouse on the pier.  The distance from the barrel (end) of the water cannon to the roof of the warehouse is 120 feet, and the water shoots up 50 feet above the barrel of the water cannon.

5. While watering her outdoor plants, Jenna noticed that the water coming out of her garden hose followed a parabolic path.  Thinking that she might be able to model the path of the water with an equation, she quickly took some measurements.  The highest point the water reached was 8 feet, and it landed on the plants 10 feet from where she was standing.  Both the nozzle of the hose and the top of the flowers were 4 feet above the ground.  Help Jenna write an equation that describes the path of the water from the hose to the top of her plants. 

6. A parabola has vertex (2, 3) and contains the point (0, 0).  Find an equation that represents this parabola.  

- Write the equation of a parabola in graphing form for each of the following

7. vertex: (2, -1) 8. vertex: (-4, 6) 9. vertex: (4, 5) point: (4, 3) point: (-1, 9) point: (8, -3)

�1010. vertex: (0, 0) 11. vertex: (1, -10) 12. vertex: (-6, -7) point: (-2, -12) point: (-3, 54) point: (0, -61)

13. vertex: (5, -2) 14. vertex: (-3, 2) 15. vertex: (6, 1) point: (4, 0) point: (-1, -18) point: (4, 5)

Circles and Generics

- Circles have two unique characteristics: a _______________ and a _______________ .

- The graphing form of a circle equation is �

1. Therefore, what is the center and radius of the following:

a. � b. � c. �

2. Write the equation of a circle with its center at (5, –7) with radius 10.

3. How would the previous circle change if I wanted to make the circle move right 10 units and down 18 units?

x − h( )2 + y − k( )2 = r2

x2 + y2 = 25 x − 3( )2 + y + 7( )2 = 169 x − 4( )2 + y +1( )2 = 16

�11- Write the equation of a circle with the given radius and whose center is the origin.

4. r = 3 5. r = 9 6. r = � 7. r = � 8. r = �

- Write the equation of a circle for the given information.

9. r = 6 10. r = 11 11. r = � 12. r = � 13. r = �vertex: (0, -10) vertex: (8, 0) vertex: (-3, -4) vertex: (-4, -1) vertex: (5, 8)

- Identify the center and radius of each.

14. � 15. � 16. �

17. � 18. � 19. �

20. � 21. �

11 5 6 3 3

7 30 4 5

x2 + y2 = 49 x2 + y2 = 324 x + 2( )2 + y − 3( )2 = 183

x + 7( )2 + y + 8( )2 = 64 x +10( )2 + y + 9( )2 = 36 x + 5( )2 + y −10( )2 = 9

x2 + y + 2( )2 = 121 x −14( )2 + y − 2( )2 = 4

�1222. � 23. �

24. � 25. �

14. Use the graph of � to sketch each graph. (Hint: graph does not need to be in center of grid.)

a. � b. � c. �

364 + 28y + y2 + x2 = −26x x2 + y2 + 24x +10y +160 = 0

−6x = −x2 + 32y − 264 − y2 −6x + x2 = 97 +10y − y2

f

y = f −x( ) y = f x( )+ 4 y = 2 f x( )

�13

d. � e. � f. �

g. �

y = − f x − 4( ) y = f x( )− 3 y = − f x( )−1

y = f 2x( )