Function Transformations: Learn and Master the Basics

Question 1

Which transformation moves a function vertically up or down while preserving its shape?

Accepted Answer

Translation

Answer

Reflection

Answer

Horizontal stretching

Answer

Vertical stretching

Question 2

What is the equation of the function that results from translating the graph of f(x) = x^2 by 2 units to the right?

Accepted Answer

f(x) = (x + 2)^2

Answer

f(x) = (x - 2)^2

Answer

f(x) = x^2 - 2

Answer

f(x) = x^2 + 2

Question 3

Which transformation creates a function that is symmetrical about the y-axis?

Accepted Answer

Reflection across the y-axis

Answer

Reflection across the x-axis

Answer

Vertical stretching

Answer

Horizontal stretching

Question 4

What happens to the x-intercepts of a function after a horizontal translation of 4 units to the left?

Accepted Answer

Shifted 4 units to the left

Answer

Shifted 4 units to the right

Answer

Remain unchanged

Answer

Move along the y-axis

Question 5

Which function represents a vertical stretch of the function f(x) by a factor of 5?

Accepted Answer

5f(x)

Answer

f(5x)

Answer

f(x) / 5

Answer

f(x) + 5

Question 6

Which of the following is the correct effect of a reflection over the y-axis on the graph of y = cos(2x)?

Accepted Answer

The graph is flipped left to right.

Answer

The graph is flipped upside down.

Answer

The graph becomes narrower.

Answer

The graph becomes wider.

Question 7

What is the effect of a horizontal compression by a factor of 2 on the graph of y = cot(x)?

Accepted Answer

The graph becomes narrower.

Answer

The graph becomes wider.

Answer

The graph is shifted to the right.

Answer

The graph is shifted to the left.

Question 8

Which of the following transformations would result in a graph of y = √x that is shifted 4 units up?

Accepted Answer

f(x) = √x + 4

Answer

f(x) = √(x + 4)

Answer

f(x) = √(x - 4)

Answer

f(x) = √x - 4

Question 9

What is the effect of a reflection over the line y = x on the graph of y = |x - 1|?

Accepted Answer

The graph is flipped left to right.

Answer

The graph is flipped upside down.

Answer

The graph becomes narrower.

Answer

The graph becomes wider.

Question 10

Which of the following transformations shifts a function vertically without changing its shape?

Accepted Answer

Translation

Answer

Reflection over the x-axis

Answer

Reflection over the y-axis

Answer

Stretching

Question 11

If f(x) = x^2 is reflected over the x-axis, what is the equation of the resulting function?

Accepted Answer

f(x) = -x^2

Answer

f(x) = x^2

Answer

f(x) = |x^2|

Answer

f(x) = -|x^2|

Question 12

What is the equation of the function g(x) if it is a vertical translation of f(x) = 2x + 1 by 3 units up?

Accepted Answer

g(x) = 2x + 4

Answer

g(x) = 2x - 2

Answer

g(x) = -2x + 4

Answer

g(x) = -2x - 2

Question 13

Which transformation would flip the graph of a function over the line y = x?

Accepted Answer

Reflection over the y-axis

Answer

Reflection over the x-axis

Answer

Rotation by 90 degrees clockwise

Answer

Rotation by 90 degrees counterclockwise

Question 14

Find the equation of the function h(x) if it is a horizontal compression of f(x) = |x| by a factor of 2.

Accepted Answer

h(x) = |x/2|

Answer

h(x) = |2x|

Answer

h(x) = 2|x|

Answer

h(x) = |x| / 2

Question 15

If f(x) = sin(x) is reflected over the y-axis and then shifted 1 unit to the left, write the equation of the resulting function.

Accepted Answer

g(x) = -sin(x + 1)

Answer

g(x) = sin(-x + 1)

Answer

g(x) = -sin(-x - 1)

Answer

g(x) = sin(x - 1)

Question 16

Which transformation results in a function that is shifted 4 units down and stretched vertically by a factor of 2?

Accepted Answer

Vertical translation of 4 units and vertical stretch of 2

Answer

Vertical stretch of 2 and vertical translation of 4 units

Answer

Horizontal translation of 4 units and vertical stretch of 2

Answer

Horizontal stretch of 2 and horizontal translation of 4 units

Question 17

Which transformation shifts the graph of the function y = x² 3 units to the right?

Accepted Answer

y = (x + 3)²

Answer

y = (x - 3)²

Answer

y = x²(x - 3)

Answer

y = x²(x + 3)

Question 18

What is the effect of reflecting the graph of the function y = |x| across the y-axis?

Accepted Answer

Flips the graph over the y-axis

Answer

Creates a V-shaped graph

Answer

Creates a U-shaped graph

Answer

Flips the graph over the x-axis

Question 19

Which transformation translates the graph of the function y = e^x 5 units to the left?

Accepted Answer

y = e^(x - 5)

Answer

y = (e^x) - 5

Answer

y = 5e^x

Answer

y = e^(x + 5)

Question 20

What is the effect of reflecting the graph of the function y = log(x) across the y-axis and then vertically stretching it by a factor of 3?

Accepted Answer

The graph is stretched vertically by a factor of 3 and inverted.

Answer

The graph is stretched vertically by a factor of 3.

Answer

The graph is inverted and the range is tripled.

Answer

The graph is inverted and the amplitude is tripled.

Question 21

What is the equation of the function obtained by stretching the graph of y = √(x + 3) vertically by a factor of 2 and then translating it 1 unit down?

Accepted Answer

y = 2√(x + 3) - 1

Answer

y = √(x + 3) - 2

Answer

y = 2√(x + 3) + 1

Answer

y = √(x + 3) + 2

Question 22

To translate the graph of the rational function y = (x+1)/(x-2) three units to the right and two units down, which transformation should you apply?

Accepted Answer

Translate right 3 units and down 2 units

Answer

Translate left 3 units and up 2 units

Answer

Reflect across the x-axis and down 2 units

Answer

Reflect across the y-axis and right 3 units

Question 23

A function f(x) is transformed to g(x) = f(x) + 3. What type of transformation is this?

Accepted Answer

Vertical translation up

Answer

Vertical translation down

Answer

Horizontal translation right

Answer

Horizontal translation left

Question 24

(Improved) A function f(x) is transformed to g(x) = f(2x + 1). What is the effect of this transformation?

Accepted Answer

Horizontal stretch by a factor of 2, and a horizontal translation 1 unit to the left

Answer

Horizontal compression by a factor of 2, and a horizontal translation 1 unit to the right

Answer

Vertical stretch by a factor of 2, and a horizontal translation 1 unit to the left

Answer

Vertical compression by a factor of 2, and a horizontal translation 1 unit to the right

Question 25

(Improved) Which transformation results in a graph that is taller and narrower?

Accepted Answer

Vertical stretch

Answer

Horizontal stretch

Answer

Vertical compression

Answer

Horizontal compression

Question 26

(Improved) A function f(x) is transformed to g(x) = f(x - 4) + 2. What type of transformation is this?

Accepted Answer

Horizontal translation 4 units to the right and a vertical translation 2 units up

Answer

Horizontal translation 4 units to the left and a vertical translation 2 units down

Answer

Vertical translation 4 units up and a horizontal translation 2 units to the right

Answer

Vertical translation 4 units down and a horizontal translation 2 units to the left

Question 27

(Improved) A function f(x) is reflected over the x-axis and then shifted 2 units to the right. What is the equation of the transformed function?

Accepted Answer

f(-x + 2)

Answer

f(x + 2)

Answer

f(-x - 2)

Answer

f(x - 2)

Question 28

(Improved) How does stretching a function by a factor of 1/2 affect its graph?

Accepted Answer

Narrower and shorter

Answer

Wider and taller

Answer

Shifted to the right

Answer

Shifted to the left

Question 29

(Improved) Which of the following functions is the result of translating the function f(x) = x^2 + 1 three units to the left?

Accepted Answer

f(x) = (x - 3)^2 + 1

Answer

f(x) = (x + 3)^2 + 1

Answer

f(x) = x^2 + 4

Answer

f(x) = x^2 - 2

Question 30

(Improved) What is the effect of reflecting a function over the y-axis and then over the x-axis?

Accepted Answer

The graph is flipped left to right and then upside down

Answer

The graph is flipped upside down and then left to right

Answer

The graph is shifted to the right and then up

Answer

The graph is shifted to the left and then down

Question 31

Which of the following transformations will move a graph downward by 2 units?

Accepted Answer

f(x) - 2

Answer

f(x) + 2

Answer

f(x) * 2

Answer

f(x / 2)

Question 32

How will compressing a graph horizontally by a factor of 1/2 affect its appearance?

Accepted Answer

Narrower and shorter

Answer

Wider and taller

Answer

Shifted to the right

Answer

Shifted to the left

Question 33

Which of the following functions is the result of reflecting the function f(x) = |x| over the y-axis?

Accepted Answer

f(x) = |-x|

Answer

f(x) = |x|

Answer

f(x) = |x + 1|

Answer

f(x) = |x - 1|

Question 34

What happens to a graph when it is shifted 4 units to the right?

Accepted Answer

Moves 4 units to the right

Answer

Moves 4 units to the left

Answer

Flips upside down

Answer

Flips left to right

Question 35

Which transformation will make a graph narrower?

Accepted Answer

f(x / 2)

Answer

f(x) + 2

Answer

f(x - 2)

Answer

f(2x)

Question 36

Which transformation shifts a function 2 units to the left and 3 units upward?

Accepted Answer

(x - 2, y + 3)

Answer

(x + 2, y - 3)

Answer

(x - 3, y + 2)

Answer

(x + 3, y - 2)

Question 37

If we reflect a function over the x-axis, what happens to its y-coordinates?

Accepted Answer

Multiply by -1

Answer

Remain unchanged

Answer

Divide by -1

Answer

Add 1

Question 38

Which expression represents a horizontal stretch of a function by a factor of 3?

Accepted Answer

f(3x)

Answer

f(x/3)

Answer

3f(x)

Answer

f(x) / 3

Question 39

Which transformation keeps the shape of a function unchanged?

Accepted Answer

Translation

Answer

Reflection

Answer

Stretching

Answer

Shrinking

Question 40

Which transformation will move the vertex of a parabola 3 units to the right and 2 units up?

Accepted Answer

Translation 3 units right and 2 units up

Answer

Reflection over the line x = 3

Answer

Vertical stretch by a factor of 2

Answer

Horizontal stretch by a factor of 3

Question 41

Which transformation does not alter the shape of a function's graph?

Accepted Answer

Translation

Answer

Reflection

Answer

Vertical stretch

Answer

Horizontal stretch

Question 42

After translating the graph of f(x) = x^2 to the right by 3 units, what is the resulting function?

Accepted Answer

f(x) = (x + 3)^2

Answer

f(x) = (x - 3)^2

Answer

f(x) = x^2 + 3

Answer

f(x) = x^2 - 3

Question 43

What is the equation of the function that is the reflection of f(x) = |x| over the x-axis?

Accepted Answer

f(x) = -|x|

Answer

f(x) = |x|

Answer

f(x) = -x

Answer

f(x) = x

Question 44

After vertically stretching the graph of f(x) = sin(x) by a factor of 2, what is the equation of the new function?

Accepted Answer

f(x) = 2sin(x)

Answer

f(x) = sin(2x)

Answer

f(x) = 1/2sin(x)

Answer

f(x) = sin(x/2)

Question 45

Which transformation will shift the graph of f(x) = x^3 upward by 4 units?

Accepted Answer

Vertical translation up 4 units

Answer

Horizontal translation up 4 units

Answer

Vertical stretch by a factor of 4

Answer

Horizontal stretch by a factor of 4

Question 46

The graph of y = f(x) is reflected across the x-axis. Which equation represents the transformed graph?

Accepted Answer

y = -f(x)

Answer

y = f(x) - 1

Answer

y = f(x) + 1

Answer

y = f(-x)

Question 47

After a vertical stretch by a factor of 2, the equation of the transformed graph of y = f(x) becomes:

Accepted Answer

y = 2f(x)

Answer

y = f(2x)

Answer

y = f(x) + 2

Answer

y = f(x) - 2

Question 48

The graph of g(x) is obtained by first vertically stretching the graph of f(x) = |x| by a factor of 3 and then shifting it 1 unit to the right. What is the equation of g(x)?

Accepted Answer

g(x) = 3|x - 1|

Answer

g(x) = |3x - 1|

Answer

g(x) = 3|x + 1|

Answer

g(x) = |3x + 1|

Question 49

Which transformation is represented by the equation f(x) + 2?

Accepted Answer

Vertical translation up 2 units

Answer

Horizontal translation left 2 units

Answer

Vertical translation down 2 units

Answer

Horizontal translation right 2 units

Question 50

What is the equation of the transformed function g(x) if it is obtained by reflecting f(x) over the x-axis?

Accepted Answer

g(x) = -f(x)

Answer

g(x) = |f(x)|

Answer

g(x) = f(-x)

Answer

g(x) = 1/f(x)

Question 51

Which transformation is represented by the graph of h(x) = f(2x)?

Accepted Answer

Horizontal stretch by a factor of 1/2

Answer

Horizontal stretch by a factor of 2

Answer

Vertical stretch by a factor of 2

Answer

Vertical stretch by a factor of 1/2

Question 52

What is the equation of the function k(x) if it is obtained by translating f(x) left 3 units?

Accepted Answer

k(x) = f(x + 3)

Answer

k(x) = f(x - 3)

Answer

k(x) = f(x) - 3

Answer

k(x) = f(3x)

Question 53

Which transformation is represented by the function j(x) = -2f(x)?

Accepted Answer

Vertical stretch by a factor of 2 and reflection over the x-axis

Answer

Vertical stretch by a factor of 2 and reflection over the y-axis

Answer

Horizontal stretch by a factor of 2 and reflection over the x-axis

Question 54

After a reflection over the y-axis followed by a vertical translation up 1 unit, the equation of the transformed function m(x) becomes:

Accepted Answer

m(x) = f(-x) + 1

Answer

m(x) = -f(-x) + 1

Answer

m(x) = f(x) - 1

Answer

m(x) = -f(x) - 1

Question 55

Which transformation is represented by the function n(x) = f(x - 2) + 3?

Accepted Answer

Horizontal translation right 2 units and vertical translation up 3 units

Answer

Horizontal translation right 2 units and vertical translation down 3 units

Answer

Horizontal translation left 2 units and vertical translation up 3 units

Question 56

What is the equation of the function p(x) if it is the result of stretching f(x) vertically by a factor of 1/3 and then reflecting it over the y-axis?

Accepted Answer

-f(3x) / 3

Answer

f(x) / 3

Answer

-f(x) / 3

Answer

f(3x) / 3

Question 57

Which transformation is represented by the graph of q(x) = -f(x + 1)?

Accepted Answer

Horizontal translation left 1 unit and reflection over the x-axis

Answer

Horizontal translation left 1 unit and reflection over the y-axis

Answer

Horizontal translation right 1 unit and reflection over the x-axis

Question 58

What is the equation of the function r(x) if it is the result of stretching f(x) horizontally by a factor of 2 and then translating it down 4 units?

Accepted Answer

f(2x) - 4

Answer

f(x/2) - 4

Answer

f(x/2) + 4

Answer

f(2x) + 4