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Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133
http://lps.lexingtonma.org/Page/2434
Name:
Date:
Function/Graph Transformation Quiz Review Students Will Be Able To: • Recognize the basic graphs including: 𝑦 = 𝑥, 𝑦 = 𝑥 ! , 𝑦 = 𝑥 ! , 𝑦 = 𝑥 , 𝑥 ! + 𝑦 ! = 1, !
𝑦 = 2! , 𝑦 = 𝑥, and 𝑦 = !. • Translate basic graphs up down, left and right and determine the resultant equation. • Reflect basic graphs over the x-­‐axis or the y-­‐axis. • Dilate (stretch or shrink) graphs by a given factor. • Compose multiple transformations. Basic Graphs: Sketch each of the basic graphs below. Show at least 3, preferably 5 specific points. 𝑦 = 𝑥 𝑦 = 𝑥 𝑦 = 𝑥 ! 𝑦 = 𝑥 ! 𝑥 ! + 𝑦 ! = 25 𝑦 = 𝑥 1
𝑦 = 𝑥
𝑦 = 2! 1 Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133
http://lps.lexingtonma.org/Page/2434
Name:
Date:
Reference Information Translations of 𝑦 = 𝑓(𝑥): (Assume that a is a positive number.) 𝑦 + 𝑎 = 𝑓 𝑥 OR 𝑦 = 𝑓 𝑥 − 𝑎 Results in the graph shifting down a units 𝑦 − 𝑎 = 𝑓 𝑥 OR 𝑦 = 𝑓 𝑥 + 𝑎 Results in the graph shifting up a units 𝑦 = 𝑓 𝑥 + 𝑎 Results in the graph shifting left a units 𝑦 = 𝑓 𝑥 − 𝑎 Results in the graph shifting right a units Reflections of 𝑦 = 𝑓(𝑥): 𝑦 = −𝑓(𝑥) Results in the graph reflecting over the x-­‐axis 𝑦 = 𝑓(−𝑥) Results in the graph reflecting over the y-­‐axis Scalings of 𝑦 = 𝑓(𝑥): !
𝑦 = 𝑎𝑓(𝑥) OR ! = 𝑓(𝑥) !
!
𝑦 = ! 𝑓(𝑥) OR 𝑎𝑦 = 𝑓(𝑥) Results in the graph becoming ! times as tall 𝑦 = 𝑓(𝑎𝑥) Results in the graph becoming ! times as wide 1
𝑥 𝑎
Results in the graph becoming a times as wide 𝑦=𝑓
Results in the graph becoming a times as tall !
2 Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133
http://lps.lexingtonma.org/Page/2434
Name:
Date:
Practice Problems: Do these problems WITHOUT YOUR CALCULATOR as you will not have that tool when you take the quiz. Directions: Each description or graph represents a different transformation or combination of transformations of some “parent” function. Fill in the missing information !
1. Equation: 2. Equation: 𝑦 = − ! 𝑥 ! + 3 Description: The graph of 𝑦 = 𝑥 is shifted Description: !
up 2 and is ! as wide. Graph: Graph: 3. Equation: Description: The graph of 𝑦 = 𝑥 ! is made twice as tall and is reflected over the y-­‐
axis. Graph: 4. Equation: 𝑦 = −4 𝑥 − 4 Description: Graph: 3 Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133
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6. Equation: 5. Equation: −𝑦 = 4𝑥 Description: The graph of 2! is reflected over the y-­‐axis and shifted down 5 units. Graph: Description: Graph: !
7. Equation: − !!! + 2 8. Equation: Description: The graph of 𝑥 ! + 𝑦 ! = 25 is made twice as tall and shifted left 4 units. Graph: Description: Graph: Name:
Date:
4 Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133
http://lps.lexingtonma.org/Page/2434
Name:
Date:
Directions: Each of the following figures is one of the basic graphs with one or more transformation performed on it. For each one, a. Identify the parent function. b. Describe in writing the transformation(s) that was/were performed. c. Write an equation for the new function. 9. 10. 8
8
6
6
4
4
2
2
–5
–5
5
–2
a. b. c. 6
11. 5
–2
a. b. c. 12. 4
4
2
2
–5
5
–5
–2
–2
–4
–6
–4
–8
a. b. c. 5
–6
a. b. c. 5 Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133
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Name:
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a. Identify the parent function. b. Describe in writing the transformation(s) that was/were performed. c. Write an equation for the new function. 13. 14. 6
2
4
–5
5
2
–2
–4
–5
–2
–6
–8
a. b. c. 5
–4
a. b. c. 15. 16. 4
6
2
–5
4
2
5
–2
–5
–4
–6
a. b. c. 5
–2
–4
a. b. c. 6 Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133
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Mixed Review from the textbook. 17. Sketch the graph of 𝑦 = 3 𝑥 − 1 + 2 Name:
Date:
18. Sketch the graph of 6
! !
!
+ (3𝑦)! = 36. 10
8
4
6
2
4
2
–5
5
– 10
–2
–5
5
10
–2
–4
–4
–6
–8
–6
Describe how your graph related to the graph of 𝑦 = 𝑥 . 19. Write an equation for a function with a graph that is the graph of 𝑦 = 𝑥 ! after a translation 3 units to the left and 1 unit up. 21. How are the graphs of 𝑦 = 𝑥 ! and 𝑦 = (𝑥 − 3)! related? 23. What does the graph of −2𝑦 = 𝑥 ! look like? – 10
How is your graph related to the graph of 𝑥 ! + 𝑦 ! = 36? 20. Write an equation for a function with a !
graph that is the graph of 𝑦 = ! made 6 times as wide. 22. What does this graph look like? (𝑥 + 1)! + (𝑦 − 4)! = 36 24. What does this graph look like? (3𝑥)! + (2𝑦)! = 36 7 Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133
http://lps.lexingtonma.org/Page/2434
Name:8
Date:
6
s(x) = – 4· x – 4
Answer Key 1. 𝑦 = 3𝑥 + 2 2. The graph is reflected over the x-­‐
6
axis, is shifted up 3 units and is half as tall. 4
6
4
3. 𝑦 = −2𝑥 ! 4
10 2
2
2
4
4.The graph is reflected over the 2x-­‐axis, shifted 4 to the right and made 4 times as tall. 6
–5
5
–2
–5
–5
5
–5
5
–2
–4
–2
–4
–4
6. 𝑦 = 2!! − 5 5. The graph is reflected over the !
x-­‐axis and made as wide. !
5
–2
2
–4
–6
–6
7. The graph is reflected over the x-­‐axis and shifted right 3 and up 2 –8
8. (𝑥 + 4)! +
! !
!
10
8
2
6
5
6
5
4
4
5
= 25 –2
2
–2
2
– 10
–4
–5
5
–2
–4
–5
–6
–6
5
10
–4
–2
–6
–8
–4
9. a. 𝑦 = 𝑥 b. the graph is shifted left three units and is twice as tall. c. 𝑦 = 2 𝑥 + 3 10. a. 𝑦 = 𝑥 ! b. The graph is 3 times as wide (or the !
graph is as tall) and is shifted up 3 c. 𝑦 =
13. a. 𝑦 = 𝑥 ! b. The graph is shifted down 4 and is twice as tall. c. 𝑦 = 2𝑥 ! − 4 17. !
units. !
!
𝑥
!
11. a. 𝑥 ! + 𝑦 ! = 25 b. The graph is shifted down 3 and right 4. c. (𝑥 − 4)! + (𝑦 + 3)! = 25 12. !
a. 𝑦 = !
b. The graph is shifted down 3 and left 6. !
c. 𝑦 =
− 3 !!!
!
+ 3 or 𝑦 = 𝑥 ! + 3 !
14. a. 𝑦 = 2! b. the graph is reflected across the x-­‐
axis and is shifted right 5 c. 𝑦 = −2!!! 18. 15. a. y = x b. the graph is reflected over the y-­‐axis, shifted down 5 units and made twice as tall (or half as wide) c. 𝑦 = −2𝑥 − 5 16. a. 𝑦 = 𝑥 b. the graph is shifted right 2 !
and is as tall. c. 𝑦 =
!
!
!
𝑥 − 2 19. 𝑦 = (𝑥 + 3)! + 1 4
6
2
4
– 10
2
–5
5
10
!
20. 𝑦 = !
–2
–5
– 10
5
–4
–2
21. The graph of 𝑦 = (𝑥 − 3)! is –4
the graph of 𝑦 = 𝑥 ! shifted 3 units to the right. –6
22. The graph is a circle with a radius of 6 and a center of (−1, 4) 23. The graph of −2𝑦 = 𝑥 ! looks like the graph of 𝑦 = 𝑥 ! reflected !
over the x-­‐axis and made as tall. !
24. the graph is a circle with a radius of 6 that is scaled to !
!
be as wide and as tall (so it !
!
actually is an ellipse, not a circle) 8 10