AP®MathsCollege BoardPrecalculusExam QuestionsPolynomial & Rational FunctionsTransformations of Functions

Transformations of Functions (College Board AP® Precalculus): Exam Questions

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Transformations of Functions

1
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1 mark

The function space f has a vertical asymptote at x equals 3, a space y-intercept at open parentheses 0 comma 5 close parentheses, a zero at x equals negative 2, and a horizontal asymptote at space y equals 1. The function g is defined by g open parentheses x close parentheses equals f open parentheses x close parentheses plus 4. Which of the following statements about the graph of g is true?

  • The graph of g has a horizontal asymptote at space y equals 1.

  • The graph of g has a vertical asymptote at x equals 7.

  • The graph of g has a space y-intercept at open parentheses 0 comma 9 close parentheses.

  • The graph of g has a zero at x equals negative 2.

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2
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1 mark

The graph of space f is shifted to the left by 5 units to produce the graph of g. Which of the following gives an expression for g \left(x\right) in terms of space f?

  • g open parentheses x close parentheses equals f open parentheses x close parentheses minus 5

  • g open parentheses x close parentheses equals f open parentheses x close parentheses plus 5

  • g open parentheses x close parentheses equals f open parentheses x minus 5 close parentheses

  • g open parentheses x close parentheses equals f open parentheses x plus 5 close parentheses

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3
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1 mark

The function g is defined by g open parentheses x close parentheses equals negative 3 f open parentheses x close parentheses.

Which of the following describes the transformation that maps the graph of space f to the graph of g?

  • A reflection over the x-axis

  • A vertical dilation by a factor of 3

  • A vertical dilation by a factor of 3 and a reflection over the x-axis

  • A horizontal dilation by a factor of 1 third and a reflection over the space y-axis

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4
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1 mark

The function space f has domain open square brackets negative 4 comma 7 close square brackets. The function g is defined by g open parentheses x close parentheses equals f open parentheses x close parentheses plus 5.

What is the domain of g?

  • open square brackets negative 20 comma 35 close square brackets

  • open square brackets negative 9 comma 2 close square brackets

  • open square brackets negative 4 comma 7 close square brackets

  • open square brackets 1 comma 12 close square brackets

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5
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1 mark

The function space f is given by space f open parentheses x close parentheses equals 2 x squared minus 3 x plus 1.

The graph of which of the following functions is the image of the graph of space f after a vertical translation of the graph of space f by 4 units?

  • space p open parentheses x close parentheses equals 2 x squared minus 3 x minus 3, because this is an additive transformation of space f that results from subtracting 4 from space f open parentheses x close parentheses.

  • q open parentheses x close parentheses equals 2 x squared plus 13 x plus 21, because this is an additive transformation of space f that results from adding 4 to each input value x.

  • r open parentheses x close parentheses equals 2 x squared minus 3 x plus 5, because this is an additive transformation of space f that results from adding 4 to space f open parentheses x close parentheses.

  • s open parentheses x close parentheses equals 2 x squared minus 19 x plus 45, because this is an additive transformation of space f that results from subtracting 4 from each input value x.

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6
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1 mark

The graph of y = f \left(x\right), consisting of three line segments and a semicircle, is shown.

Graph of a piecewise function f on an axes from −6 ≤ x ≤ 6, consisting of a line segment from (−4, 2) to (−2, 0), a semicircle from (−2, 0) curving down to a minimum at (0, −2) and back up to (2, 0), and a line segment from (2, 0) to (4, 3). Closed dots at the endpoints (−4, 2) and (4, 3).
Graph of f

Which of the following is the transformed graph for y = f \left(x - 2\right) + 3 ?

  • Graph of a piecewise function on axes from −6 to 6 (x) and −6 to 6 (y). A line segment from (−2, 5) descends to (0, 3), a semicircle curves down from (0, 3) to a minimum at (2, 1) and back up to (4, 3), and a line segment rises steeply from (4, 3) upward. Closed dot at (−2, 5).
  • Graph of a piecewise function on axes from −6 to 6 (x) and −6 to 6 (y). A line segment descends from upper left to (−4, 3), a semicircle curves down from (−4, 3) to a minimum at (−2, 1) and back up to (0, 3), and a line segment rises steeply from (0, 3) upward.
  • Graph of a piecewise function on axes from −6 to 6 (x) and −6 to 6 (y). A line segment from (−2, −1) descends to (0, −3), a semicircle curves down from (0, −3) to a minimum at (2, −5) and back up to (4, −3), and a line segment rises from (4, −3) to (6, 0). Closed dots at (−2, −1) and (6, 0).
  • Graph of a piecewise function on axes from −6 to 6 (x) and −6 to 6 (y). A line segment descends from upper left to (−4, −3), a semicircle curves down from (−4, −3) to a minimum at (−2, −5) and back up to (0, −3), and a line segment rises from (0, −3) to (2, 0). Closed dot at (2, 0).

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7
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1 mark

The function f is given by f \left(x\right) = 2 x^{2} - x + 4. The graph of which of the following functions is the image of the graph of f after a vertical dilation of the graph of f by a factor of 3 ?

  • m \left(x\right) = 18 x^{2} - 3 x + 4, because this is a multiplicative transformation of f that results from multiplying each input value x by 3.

  • k \left(x\right) = 6 x^{2} - 3 x + 12, because this is a multiplicative transformation of f that results from multiplying f \left(x\right) by 3.

  • p \left(x\right) = 2 \left(x + 3\right)^{2} - \left(x + 3\right) + 4, because this is an additive transformation of f that results from adding 3 to each input value x.

  • n \left(x\right) = 2 x^{2} - x + 7, because this is an additive transformation of f that results from adding 3 to f \left(x\right).

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8
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1 mark

The functions f and g are defined for all real numbers such that g \left(x\right) = f \left(3 \left(x + 2\right)\right). Which of the following sequences of transformations maps the graph of f to the graph of g in the same x y-plane?

  • A horizontal dilation of the graph of f by a factor of 3, followed by a horizontal translation of the graph of f by - 2 units

  • A horizontal dilation of the graph of f by a factor of 3, followed by a horizontal translation of the graph of f by 2 units

  • A horizontal dilation of the graph of f by a factor of \frac{1}{3}, followed by a horizontal translation of the graph of f by - 2 units

  • A horizontal dilation of the graph of f by a factor of \frac{1}{3}, followed by a horizontal translation of the graph of f by 2 units

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9
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1 mark

x

-16

-8

-6

-4

-2

0

8

f(x)

55

31

22

6

-8

-11

12

The table gives values for a polynomial function f at selected values of x. Let g(x) = af(bx) + c, where a, b, and c are positive constants. In the xy-plane, the graph of g is constructed by applying three transformations to the graph of f in this order: a horizontal dilation by a factor of 2, a vertical dilation by a factor of 2, and a vertical translation by 3 units upward. What is the value of g(-8)?

  • 12

  • 15

  • 65

  • 113

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10
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1 mark

The function g is defined by g open parentheses x close parentheses equals f open parentheses 2 x close parentheses. The graph of g is the graph of space f after which of the following transformations?

  • a horizontal dilation of the graph of space f by a factor of \frac{1}{2}

  • a horizontal dilation of the graph of space f by a factor of 2

  • a vertical dilation of the graph of space f by a factor of \frac{1}{2}

  • a vertical dilation of the graph of space f by a factor of 2

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11
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1 mark

The function g is defined by g open parentheses x close parentheses equals f open parentheses negative x close parentheses. The graph of g is the graph of space f after a reflection over which of the following?

  • the line space y equals x

  • the origin

  • the x-axis

  • the space y-axis

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12
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1 mark

The function space f has domain open square brackets negative 3 comma 5 close square brackets and range open square brackets 1 comma 8 close square brackets. The function g is defined by g open parentheses x close parentheses equals negative 2 f open parentheses x plus 1 close parentheses plus 3. What is the range of g?

  • open square brackets negative 16 comma negative 2 close square brackets

  • open square brackets negative 13 comma 1 close square brackets

  • open square brackets negative 1 comma 16 close square brackets

  • open square brackets 5 comma 19 close square brackets

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13
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1 mark

Let space f open parentheses x close parentheses equals square root of x. The function g is defined by g open parentheses x close parentheses equals negative 2 f open parentheses x minus 1 close parentheses plus 5. What is the value of g open parentheses 8 close parentheses?

  • negative 5.292

  • negative 1.000

  • negative 0.292

  • 10.292

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14
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1 mark

The function space f has a relative minimum at the point open parentheses 3 comma negative 2 close parentheses. The function g is defined by g open parentheses x close parentheses equals 1 half f open parentheses x close parentheses.

Which of the following are the coordinates of the relative minimum of the graph of g?

  • open parentheses 3 over 2 comma negative 2 close parentheses

  • open parentheses 3 comma negative 2 close parentheses

  • open parentheses 3 comma negative 1 close parentheses

  • open parentheses 3 comma 1 close parentheses

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15
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1 mark

The function g is defined by g open parentheses x close parentheses equals 3 f open parentheses 2 x minus 8 close parentheses minus 5. Which of the following describes the horizontal transformations that take the graph of f to the graph of g?

  • A horizontal dilation by a factor of 1 half, and a horizontal translation to the right by 4 units.

  • A horizontal dilation by a factor of 1 half, and a horizontal translation to the right by 8 units.

  • A horizontal dilation by a factor of 2, and a horizontal translation to the right by 4 units.

  • A horizontal dilation by a factor of 2, and a horizontal translation to the right by 8 units.

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16
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1 mark

The table gives values of a function space f at selected values of x.

space x

0

2

4

space f open parentheses x close parentheses

10

5

- 1

The function g is defined by g open parentheses x close parentheses equals negative 3 f open parentheses 1 half open parentheses x minus 4 close parentheses close parentheses plus 7.

What is the value of g open parentheses 8 close parentheses?

  • - 23

  • - 15

  • - 8

  • 22

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17
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1 mark

The function f has domain open square brackets negative 6 comma 12 close square brackets. The function g is defined by g open parentheses x close parentheses equals f open parentheses negative 2 x plus 4 close parentheses.

What is the domain of g?

  • open square brackets negative 10 comma negative 1 close square brackets

  • open square brackets negative 10 comma 8 close square brackets

  • open square brackets negative 5 comma 4 close square brackets

  • open square brackets negative 4 comma 5 close square brackets

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