Name: ________________________ Date: _________
T-1: Parent Function Graphs
Give the name of each parent function. Then make a table of values and graph each parent function.
1. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥
22. 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 3. 𝑓𝑓(𝑥𝑥) = ln 𝑥𝑥
4. 𝑓𝑓(𝑥𝑥) = |𝑥𝑥| 5. 𝑓𝑓(𝑥𝑥) = 2
𝑥𝑥6. 𝑓𝑓(𝑥𝑥) =
𝑥𝑥1Make a table of values then graph each function. Write the equation of the parent function this graph belongs to.
7. 𝑓𝑓(𝑥𝑥) = 2|𝑥𝑥 + 3| 8. 𝑓𝑓(𝑥𝑥) = log(𝑥𝑥 − 2) 9. 𝑓𝑓(𝑥𝑥) = −2(𝑥𝑥 − 3)
2+ 1 10. 𝑓𝑓(𝑥𝑥) =
𝑥𝑥−1−111. 𝑓𝑓(𝑥𝑥) = 6 × 3
−𝑥𝑥+ 1 12. 𝑓𝑓(𝑥𝑥) = �2(𝑥𝑥 − 3) + 2
Give the name of the parent function that matches each graph.
13. 14.
15. 16.
17. 18.
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Worksheet by Kuta Software LLC
Honor's Algebra 2
T-2: Translation of Graphs
Name___________________________________
Date________________ Period____
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Sketch the graph of each function. Then describe in words the transformation.
1) y = x + 3 - 4 2) y = x + 1
3) f ( x ) = 3
x + 2+ 1 4) f ( x ) = log ( x + 1 ) - 1
5) y = x - 1 - 3 6) y = x - 3 + 1
7) f ( x ) = 4
x - 1 + 1 8) f (x) = ( x + 4 )2 + 3
9) y = x + 1 - 2 10) f ( x ) = log ( x - 2 ) - 2
11) y = -5 + x + 3 12) y = x + 5 - 2
13) f ( x ) = 1
x - 1 - 3 14) y = x − 3
15) f ( x ) = 2
x - 2- 2 16) f ( x ) = ln ( x + 2 ) - 3
17) f ( x ) = 1
x + 2 - 1 18) f (x) = ( x + 1 )2 - 3
- 1
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Worksheet by Kuta Software LLC
Honor's Algebra 2
T-3: The Stretching of Graphs
Name___________________________________
Date________________ Period____
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Sketch the graph of each function. Then describe in words the transformation.
1) y = 3 2x - 4 - 1
2) f (x) = 1 3
x2+ 2
3) f (x) = 1 4
× 2
x - 2- 1 4) y = -1 + 3
x + 15) y = -5 + 5
x - 26) f (x) = 4
x + 1- 1
7) y = ln ( 4x - 2 ) + 2 8) y = log ( 3x + 9 ) - 5
9) y = 3 3x - 4 10) y = 3 3x - 2 + 3
11) y = 3
x - 112) f (x) = 4
x - 2+ 2
13) f (x) = (x + 6)
2 - 1 14) f (x) = (x - 4)
2
)
215) f (x) = 5 × 2
x - 2 + 2 16) f (x) = 3 × 2
x - 1+ 2
17) f (x) = 3
x - 2
- 2 18) y = log ( 4x + 8 ) - 1
Name: ________________________ Date: _________
T-5: Combining Stretching and More Transformations
Write an equation for each function with the following transformations. Write as one vertical stretch when appropriate. Eliminate y-axis reflection when appropriate.
1. A quadratic function that has been: 2. A base two exponential function that has been:
Reflected over the x-axis. Stretched horizontally by a scale factor of . Then reflected over the y-axis. Then reflected over the x-axis
Then shifted up 2 and left 3. Then shifted down 1 and left 2.
3. An absolute value function that has been: 4. A reciprocal function that has been:
Stretched vertically by a scale factor of 2. Stretched horizontally by a scale factor of 2.
Then reflected over the y-axis. Then reflected over the x-axis.
Then reflected over the x-axis. Then stretched vertically by a scale factor of 2.
Then translated up 4. Then translated right 3.
5. An absolute value function that has been: 6. A square root function that has been:
Stretched vertically by a scale factor of 4. Reflected over the y-axis.
Then stretched horizontally by a scale factor of . Then stretched horizontally by a scale factor of .
Then translated up 2. Then translated left 3.
7. A quadratic function that has been: 8. A square root function that has been:
Stretched horizontally by a scale factor of 2. Reflected over the x-axis.
Then stretched vertically by a scale factor of 4. Then stretched vertically by a scale factor of 3.
Then translated left 3. Then stretched horizontally by a scale factor of . Then translated down 1.
9. A reciprocal function that has been: 10. A quadratic function that has been:
Stretched horizontally by a scale factor of 3. Reflected over the x-axis.
Then stretched vertically by a scale factor of 2. Then reflected over the y-axis.
Then translated up 7 and left 4. Then translated right 3 and down 2.
11. An absolute value function that has been: 12. A square root function that has been:
Stretched horizontally by a scale factor of . Stretched vertically by a scale factor of .
Then stretched vertically by a scale factor of 5. Then stretched horizontally by a scale factor of . Then reflected over the y-axis. Then reflected over the x-axis.
Then reflected over the x-axis. Then reflected over the y-axis.
Then translated left 2 and down 1. Then translated right 12 and down 3.
Name: ________________________ Date: _________
T-6: Making Equations from Graphs
Write an equation for each graph.
1. 2.
3. 4.
Graph the following equations on the same set of axes.
5. 6.
4 3 2 2√ 2 1
2 6 2 √4 8 1
7. 8.
| 3 3| 2 3
3| 1| 2 3
Name: ________________________ Date: _________
T-7: Transformation Order
Write a simplified equation for each function with the following transformations. Be careful, the transformations are often done in a more confusing order.
1. A quadratic function that has been: 2. A base two exponential function that has been:
Reflected over the x-axis. Stretched horizontally by a scale factor of . Then shifted up 2 and left 3. Then shifted down 1 and left 2.
3. A square root function that has been: 4. A reciprocal function that has been:
Stretched vertically by a scale factor of 2. Stretched horizontally by a scale factor of 2.
Then translated up 4 and right 2 Then translated right 3 and down 2.
Then reflected over the y-axis. Then reflected over the x-axis.
5. A reciprocal function that has been: 6. A natural log function that has been:
Translated up 7 and left 4. Translated right 3 and down 2.
Then stretched vertically by a scale factor of 2. Then reflected over the y-axis.
Then stretched horizontally by a scale factor of 3. Then reflected over the x-axis.
7. An absolute value function that has been: 8. A square root function that has been:
Translated left 2 and down 1. Translated right 12 and down 3.
Then stretched horizontally by a scale factor of . Then stretched vertically by a scale factor of . Then stretched vertically by a scale factor of 3. Then reflected over the x-axis.
Write a simplified equation that results when the following transformations are applied to the given parent function.
9. Parent function: : a. Shift right 7 units,
then reflect over the x-axis,
then stretch vertically by a factor of 5, then shift upward 1 unit.
b. Reflect over the x-axis, then shift right 7 units,
then stretch vertically by a factor of 5, then shift upward 1 unit.
c. Stretch vertically by a factor of 5, then shift upward 1 unit,
then shift right 7 units, then reflect over the x-axis.
d. Shift right 7 units, then shift upward 1 unit, then reflect over the x-axis,
then stretch vertically by a factor of 5.
e. Reflect over the x-axis, then shift right 7 units, then shift upward 1 unit,
then stretch vertically by a factor of 5 .
Name: ________________________ Date: _________
T-8: Transformation of New Parent Equations
NEW PARENT FUNCTIONS
Circle Sine Cosine
𝑥
2+ 𝑦
2= 1
2𝑓(𝑥) = sin 𝑥 𝑓(𝑥) = 𝑐𝑜𝑠 𝑥
Hyperbola Cubic Cube Root
𝑥
2− 𝑦
2= 1 𝑓(𝑥) = 𝑥
3𝑓(𝑥) = √𝑥
3Make a sketch of each equation
1. 𝑓(𝑥) = 2(𝑥 − 3)
3− 2 2. 𝑓(𝑥) = 3 √𝑥 − 2
3+ 2
3. (𝑥 − 1)
2+ (𝑦 + 1)
2= 3
24. (𝑥 + 2)
2− (𝑦 + 3)
2= 1
5. (𝑥 − 1)
2− (𝑦 − 3)
2= 1 6. (𝑥 + 2)
2+ 𝑦
2= 5
2Name: ________________________ Date: _________
T-8: Transformation of New Parent Equations
Use the following coordinate grid as a template as you graph each of the equations below.
7. 𝑓(𝑥) = 3 sin 𝑥 8. 𝑓(𝑥) =
12
sin (2𝑥 − 180)
9. 𝑓(𝑥) = −𝑐𝑜𝑠 𝑥 10. 𝑓(𝑥) = 2 cos(𝑥 − 180) − 1
Make a sketch of each equation
11. Given 𝑥
2+ 𝑦
2= 1 , translate up 2, right 5 and dilate by a scale factor of 2.
12. Given 𝑥
2+ 𝑦
2= 1 , translate down 3, stretch vertically by a scale factor of 2, and stretch horizontally by a scale factor of 1/2. What is the name of this new transformed circle?
Describe the transformation. Do not sketch.
13. 𝑓(𝑥) = √𝑥 + 1
3− 4
14. 𝑓(𝑥) = 3(𝑥 − 2)
3+ 1
15. (x – 2)
2+ (y + 3)
2= 16
16. (x + 4)
2– (y – 3)
2= 1
Name: ________________________ Date: _________
T-9: More Transformation
Suppose that a student looks at a transformation of y = f(x) and breaks it into the following steps. State the transformation that occurs in each step below:
1. 𝑦 = 2𝑓(−𝑥 − 3) + 4
a. From: 𝑦 = 𝑓(𝑥) To: 𝑦 = 𝑓(−𝑥) b. From: 𝑦 = 𝑓(−𝑥) To: 𝑦 = 2𝑓(−𝑥)
c. From: 𝑦 = 2𝑓(−𝑥) To: 𝑦 = 2𝑓(−(𝑥 + 3)) = 2𝑓(−𝑥 − 3) d. From: 𝑦 = 2𝑓(−𝑥 − 3) To: 𝑦 = 2𝑓(−𝑥 − 3) + 4
2. 𝑦 = −
14
𝑓(1 − 𝑥) − 5 *Notice that (1 – x) is equivalent to − (x − 1) a. From: 𝑦 = 𝑓(𝑥) To: 𝑦 = 𝑓(−𝑥)
b. From: 𝑦 = 𝑓(−𝑥) To: 𝑦 = −𝑓(−𝑥) c. From: 𝑦 = −𝑓(−𝑥) To: 𝑦 = −
14
𝑓(−𝑥) d. From: y = −
14
𝑓(−𝑥) To: : 𝑦 = −
14
𝑓(−(𝑥 − 1)) = −
14
𝑓(−𝑥 + 1) = −
14
𝑓(1 − 𝑥) e. From: : 𝑦 = −
14
𝑓(1 − 𝑥) To: : 𝑦 = −
14
𝑓(1 − 𝑥) − 5
Describe how the graph of g is obtained from the graph of f. (Do not sketch the graph.)
3. 𝑓(𝑥) = √𝑥 , 𝑔(𝑥) = √−𝑥 − 2 4. 𝑓(𝑥) = 𝑥
3, 𝑔(𝑥) = −2(𝑥 + 5)
35. 𝑓(𝑥) = |𝑥| , 𝑔(𝑥) = −5|𝑥 − 2| + 1 6. 𝑓(𝑥) = 𝑥
2, 𝑔(𝑥) =
16
(𝑥 + 3)
2− 7 7. 𝑓(𝑥) =
1𝑥
, 𝑔(𝑥) =
3𝑥+8
+ 2 8. 𝑓(𝑥) = √𝑥
3, 𝑔(𝑥) = √−𝑥
3+ 4 9. 𝑓(𝑥) = 2
𝑥𝑔(𝑥) = −2
𝑥−210. 𝑓(𝑥) = 2
𝑥𝑔(𝑥) = 2
2𝑥+ 2 11. 𝑓(𝑥) = 2
𝑥𝑔(𝑥) = 2
−0.5𝑥12. 𝑓(𝑥) = sin 𝑥 𝑔(𝑥) = −2sin(3𝑥 + 270) + 1
Name: ________________________ Date: _________
T-9: More Transformation
Make a sketch of each function
13) 𝑓(𝑥) = −2(𝑥 + 3)
3− 1 14) 𝑓(𝑥) =
34
|𝑥 − 4| + 2
15. 𝑓(𝑥) = −(𝑥 + 3)
2+ 4 16. 𝑓(𝑥) = −2
𝑥+ 6
17. 𝑓(𝑥) = 3(𝑥 − 4)
2− 2 18. 𝑓(𝑥) =
12
√−𝑥 + 1
19. 𝑓(𝑥) = −2|𝑥 + 3| + 1 20. 𝑓(𝑥) = ln(𝑥 + 1) − 2
Identify the equation of each function.
21. 22.
23. 24.
Name: ________________________ Date: _________
T-10: Transformation Practice Test
1. Use the Parent function 𝑦 =
15𝑥
2to describe each of the following transformations.
a.
𝑦 =
15
𝑥
2+ 7 b. 𝑦 = (𝑥 − 5)
2c. 𝑦 =
−15
(3𝑥)
2− 4
2. Given 𝑦 = 2 sin(𝑥) + 7:a. Write an equation whose graph is shifted 3 units to the right.
b. Write an equation whose graph is shifted 3 units up.
c. Write an equation whose graph is reflected over the x-axis and stretched horizontally by a scale factor of 13 3. Write an equation for each graph below. The parent function is
𝑦 = 𝑥
2.
Graph A Graph B Graph C Graph D
4. Use the two given functions to choose the best statement comparing the graphs of each other.
Function 1: 𝑦 = 2
𝑥+ 4 Function 2: 𝑦 = 2
𝑥+5− 2
A. Function 2’s graph is shifted up 6 units and right 5 units from Function 1’s graph.
B. Function 2’s graph is shifted up 5 units and down 2 units from Function 1’s graph.
C. Function 2’s graph is shifted right 5 units and down 6 units from Function 1’s graph.
D. Function 2’s graph is shifted left 5 units and down 6 units from Function 1’s graph.
5. Given
y = 2 | x | +5 , write an equation whose graph is reflected, shifted to the left 2 units and shifted up 4 units.6. Given Function 1:
𝑦 = (𝑥 − 3)
2+ 1 and Function 2: 𝑦 = (𝑥 + 2)
2− 3:
a. Describe the horizontal and vertical shifts from the graph of Function 1 to the graph of Function 2.
b. Write an equation for Function 3 whose graph is only shifted right 2 and up 3 from Function 2’s.
Name: ________________________ Date: _________
T-10: Transformation Practice Test
7. Given 𝑦 = 3 cos(𝑥) − 7, write an equation whose graph is reflected over the x-axis and is twice as tall.
8. Given Function 1: 𝑦 = 3|𝑥 − 5| + 1 and Function 2: 𝑦 = −1|𝑥 + 2| − 2:
a. Describe the horizontal and vertical shifts from the graph of Function 1 to the graph of Function 2.
b. Write an equation for Function 3 whose graph is shifted right 1 and down 2, then reflected over the x-axis from Function 2’s. (Pay close attention to the order in which the transformations are done.)
9. Given f (x) = 2x , write an equation whose graph is shifted to the right 5 units, up 3 units, then reflected over the x- axis. (Be careful.)
Identify the parent function whose equation is given, describe the transformation and graph.
10. 𝑓(𝑥) = log(2𝑥 + 14) − 3 11. 𝑓(𝑥) = −√3𝑥 − 27 12. 𝑓(𝑥) = −13𝑥4+ 3
Simplify the transformation, then describe.
13. 𝑓(𝑥) = −15(−2𝑥)2+ 5 14. 𝑓(𝑥) = −2(−2𝑥)3− 2 15. 𝑓(𝑥) = −3𝑥−96 + 2
For questions #16-19, write an equation in standard form that has a graph with the given characteristics.
16. The shape of 𝑦 = 𝑥2, but reflected over the x-axis and then shifted left 3 units.
17. The shape of 𝑦 = √𝑥, but reflected over the y-axis and then shifted right 2 units and down 1 18. The shape of 𝑦 =1𝑥, but reflected over the x axis, vertical stretch by a scale factor of 3, and then shifted left 1 and shifted up 6 units.
19. The shape of 𝑦 = |𝑥|, but shifted right 6 and down 4 units, then horizontal stretch by scale factor of 1/2, then reflected over the x axis. (Be Careful)
20. Use the two given functions to choose the best statement comparing the graphs to each other.
