Function Notation and Transformations Practice Questions — SAT

Question 1Question →

The graph of the function $y = f(x)$ in the $xy$ -plane passes through the point $(3, -4)$ . If the function $g$ is defined by $g(x) = f(x + 2) - 1$ , which of the following points must lie on the graph of $y = g(x)$ ?

(1, -5)

(5, -5)

(1, -3)

(5, -3)

Show answer & explanation

Answer: $(1, -5)$

Answer

(1, -5)

The correct answer is the point

(1, -5)

. Since the point

(3, -4)

lies on the graph of

f

, we know that

f(3) = -4

. For the function

g(x) = f(x + 2) - 1

, substituting

x = 1

gives

g(1) = f(1 + 2) - 1 = f(3) - 1

. Substituting

-4

for

f(3)

yields

g(1) = -4 - 1 = -5

. Thus, the point

(1, -5)

must lie on the graph of

g

.

Step-by-Step Solution

1

Identify the given function value from the point on the graph of

f

.

Since the graph of

y = f(x)

passes through

(3, -4)

, we have

f(3) = -4

.

Any point

(x, y)

on the graph of a function satisfies the equation

y = f(x)

.

2

Determine the input variable

x

for the function

g(x) = f(x + 2) - 1

that corresponds to the known input of

3

for

f

.

Set the argument of

f

in

g(x)

equal to

3

:

x + 2 = 3

, which simplifies to

x = 1

.

This allows us to substitute the known value

f(3)

into the expression for

g(x)

.

3

Evaluate

g(1)

using the value of

f(3)

.

g(1) = f(1 + 2) - 1 = f(3) - 1 = -4 - 1 = -5

.

Substituting

f(3) = -4

into the simplified expression gives the output value of

g

at

x = 1

.

4

State the resulting point on the graph of

y = g(x)

.

The point is

(1, -5)

.

An input of

x = 1

yields an output of

y = -5

for the function

g

.

Key Concept

Function transformations and their effects on individual coordinate points.

Question 2Question →

The graph of the function $f$ in the $xy$ -plane is translated $4$ units to the right and $3$ units down to create the graph of the function $g$ . Which of the following equations defines $g(x)$ in terms of $f(x)$ ?

g(x) = f(x + 4) - 3

g(x) = f(x - 4) + 3

g(x) = f(x - 4) - 3

g(x) = f(x + 4) + 3

Show answer & explanation

Answer: $g(x) = f(x - 4) - 3$

Answer

The equation that defines

g(x)

in terms of

f(x)

is

g(x) = f(x - 4) - 3

.

The equation

g(x) = f(x - 4) - 3

correctly represents the translated function. A translation of

4

units to the right is represented by replacing

x

with

x - 4

within the function's input, and a translation of

3

units down is represented by subtracting

3

from the outer function.

Step-by-Step Solution

1

Apply the horizontal translation to the function input.

f(x - 4)

Translating a function

f(x)

horizontally to the right by

h

units is represented by substituting

x

with

x - h

. For a shift of

4

units to the right, we replace

x

with

x - 4

.

2

Apply the vertical translation to the function output.

f(x - 4) - 3

Translating a function vertically down by

k

units is represented by subtracting

k

from the entire function. For a shift of

3

units down, we subtract

3

.

3

Write the final equation for

g(x)

.

g(x) = f(x - 4) - 3

Combining the horizontal and vertical transformations yields the function

g(x)

.

Key Concept

Graph transformations of functions

Question 3Question →

The function $f$ has the property that $f(3) = 11$ . The function $g$ is defined by $g(x) = f(x) - 4$ . What is the value of $g(3)$ ?

Show answer & explanation

Answer: 7

Answer

7

The value of

g(3)

is found by substituting

x = 3

into the equation

g(x) = f(x) - 4

, which results in

g(3) = f(3) - 4

. Substituting the given value of

f(3) = 11

yields

g(3) = 11 - 4

, which simplifies to

7

.

Step-by-Step Solution

1

Substitute

x = 3

into the function definition of

g(x)

.

g(3) = f(3) - 4

To evaluate the function

g

at a specific input, we replace

x

with

3

in the definition

g(x) = f(x) - 4

.

2

Substitute the given value of

f(3) = 11

into the equation.

g(3) = 11 - 4

The problem states that the value of

f(3)

is equal to

11

.

3

Simplify the expression to find the final value.

g(3) = 7

Subtracting

4

from

11

yields

7

.

Key Concept

Applying vertical translations using function notation.

Estimated Time:40s

Question 4Question →

The table below shows several values of the function $f$ .

$x$	$f(x)$
$-3$	$8$
$-1$	$2$
$1$	$5$
$3$	$-1$

The function $g$ is defined by $g(x) = f(x + 2)$ . What is the value of $g(-1)$ ?

8

5

4

0

Show answer & explanation

Answer: 5

Answer

5

To find the value of

g(-1)

, substitute

x = -1

into the equation

g(x) = f(x + 2)

, which yields

g(-1) = f(-1 + 2) = f(1)

. Looking at the table, when the input is

1

, the output of the function

f

is

5

. Therefore, the value of

g(-1)

is

5

.

Step-by-Step Solution

1

Substitute the given input value of

-1

for

x

in the definition of

g(x)

.

g(-1) = f(-1 + 2)

To find

g(-1)

, we need to replace

x

with

-1

in the equation

g(x) = f(x + 2)

.

2

Simplify the expression inside the function notation.

g(-1) = f(1)

Performing the addition

-1 + 2

yields

1

.

3

Use the table to find the value of

f(1)

.

f(1) = 5

Looking at the row in the table where

x = 1

, the corresponding value of

f(x)

is

5

.

Key Concept

Function transformations and notation using tables

Question 5Question →

For the function $f$ , it is given that $f(4) = 18$ . The function $g$ is defined by $g(x) = \frac{1}{3}f(x + 2)$ . What is the value of $g(2)$ ?

Show answer & explanation

Answer: 6

Answer

The value of

g(2)

is 6.

To evaluate

g(2)

, substitute

x = 2

into the definition

g(x) = \frac{1}{3}f(x+2)

, which yields

g(2) = \frac{1}{3}f(2+2) = \frac{1}{3}f(4)

. Since

f(4) = 18

, this simplifies to

\frac{1}{3}(18) = 6

.

Step-by-Step Solution

1

Substitute

x = 2

into the expression for

g(x)

.

g(2) = \frac{1}{3}f(2 + 2)

To evaluate the function

g

at

x = 2

, we substitute 2 for every occurrence of

x

in the function definition.

2

Simplify the input argument for the function

f

.

g(2) = \frac{1}{3}f(4)

Adding 2 and 2 inside the parentheses simplifies the input of

f

to 4.

3

Substitute the given value of

f(4)

into the simplified expression.

g(2) = \frac{1}{3}(18)

The problem states that

f(4) = 18

.

4

Perform the final multiplication.

6

Multiplying 18 by

\frac{1}{3}

is equivalent to dividing 18 by 3, which yields 6.

Key Concept

Evaluating a transformed function by substituting a value into function notation.

Estimated Time:45s

Question 6Question →

The graph of the function $f$ in the $xy$ -plane passes through the point $(5, -2)$ . The function $g$ is defined by $g(x) = f(x) + 6$ . Which of the following points must lie on the graph of $g$ ?

(5, -8)

(11, -2)

(5, 4)

(-1, -2)

Show answer & explanation

Answer: $(5, 4)$

Answer

The point

(5, 4)

Since the point

(5, -2)

is on the graph of

f

, we know that

f(5) = -2

. The function

g

is defined by

g(x) = f(x) + 6

, which represents a vertical shift upward by

6

units. Substituting

x = 5

into the equation for

g

yields

g(5) = f(5) + 6 = -2 + 6 = 4

. Therefore, the point

(5, 4)

must be on the graph of

g

.

Step-by-Step Solution

1

Determine the value of

f(5)

using the given point on the graph of

f

.

f(5) = -2

Since the point

(5, -2)

lies on the graph of

f

, the input

x = 5

corresponds to the output

y = -2

.

2

Use the definition of

g(x)

to evaluate

g(5)

.

g(5) = f(5) + 6 = -2 + 6 = 4

Substitute

x = 5

into the definition

g(x) = f(x) + 6

to find the corresponding

y

-value for the graph of

g

.

Key Concept

Vertical translations of function graphs

Estimated Time:45s

Question 7Question →

If the graph of $y = f(x)$ contains the point $(3, 7)$ , and the function $g$ is defined by $g(x) = f(x + 4) - 2$ , what is the value of $g(-1)$ ?

Show answer & explanation

Answer: 5

Answer

5

Since the graph of

y = f(x)

contains the point

(3, 7)

, we have

f(3) = 7

. The function

g

is defined as

g(x) = f(x + 4) - 2

. To find the value of

g(-1)

, we substitute

x = -1

into the definition of

g

:

g(-1) = f(-1 + 4) - 2 = f(3) - 2

. Substituting

f(3) = 7

gives

g(-1) = 7 - 2 = 5

.

Step-by-Step Solution

1

Translate the point

(3, 7)

on the graph of

f(x)

into function notation.

f(3) = 7

By definition, if a point

(a, b)

is on the graph of

y = f(x)

, then

f(a) = b

.

2

Substitute

x = -1

into the expression for

g(x)

to evaluate

g(-1)

.

g(-1) = f(-1 + 4) - 2

To find

g(-1)

, replace every occurrence of

x

with

-1

in the function definition of

g(x)

.

3

Simplify the input of the function

f

and compute the final value.

g(-1) = f(3) - 2 = 7 - 2 = 5

Simplify

-1 + 4

to

3

, then substitute the known value

f(3) = 7

and subtract

2

.

Key Concept

Evaluating a transformed function using function notation and given coordinate points.

Question 8Question →

The function $f$ is defined for all real numbers, and the graph of $y = f(x)$ in the $xy$ -plane has a single minimum at the point $(5, -2)$ . The function $g$ is defined by $g(x) = -3f(2x - 4) + 7$ . What is the $y$ -coordinate of the maximum point on the graph of $y = g(x)$ ?

Show answer & explanation

Answer: 13

Answer

The correct answer is 13.

The graph of

y = f(x)

has a minimum at

(5, -2)

, which means

f(5) = -2

and

f(x) \ge -2

for all

x

. The function

g(x) = -3f(2x-4) + 7

includes a vertical stretch by a factor of

3

, a vertical reflection across the

x

-axis, and a vertical shift upward by

7

units. Because of the vertical reflection, the minimum value of the original function becomes the maximum value of the transformed function. Applying the vertical transformations to the

y

-coordinate of the minimum point yields

-3(-2) + 7 = 6 + 7 = 13

.

Step-by-Step Solution

1

Identify the minimum point and minimum value of the original function

f(x)

.

f(5) = -2

, and

f(x) \ge -2

for all real numbers

x

.

The problem states that the graph of

y = f(x)

has a single minimum at the point

(5, -2)

.

2

Determine the transformed

x

-coordinate corresponding to the original input of

5

.

2x - 4 = 5 \implies 2x = 9 \implies x = 4.5

.

Setting the argument of the function

f(2x-4)

equal to the original minimum input of

5

allows us to find the corresponding input

x

for the function

g

.

3

Apply the vertical transformations to find the output value of

g(x)

at

x = 4.5

.

g(4.5) = -3f(5) + 7 = -3(-2) + 7 = 6 + 7 = 13

.

Substituting

f(5) = -2

into the definition of

g(x)

gives the vertical transformation of the point.

4

Confirm that the point is indeed the maximum of the transformed function

g(x)

.

Since

f(2x-4) \ge -2

, multiplying by

-3

yields

-3f(2x-4) \le 6

. Adding

7

yields

g(x) \le 13

, confirming that

13

is the maximum value.

Multiplying a function by a negative number reflects its graph vertically, changing a minimum point into a maximum point.

Key Concept

Applying horizontal and vertical transformations to function coordinates, and understanding how vertical reflections affect the extrema (minima and maxima) of a graph.

Question 9Question →

The function $f$ is defined by $f(x) = 2x - 3$ . If the function $g$ is defined by $g(x) = f(x + 4)$ , what is the value of $g(1)$ ?

-9

3

7

13

Show answer & explanation

Answer: 7

Answer

The correct answer is 7.

Since

g(x) = f(x + 4)

, evaluating

g(1)

requires finding

f(1 + 4)

, which is

f(5)

. Substituting

5

into the expression for

f(x)

gives

f(5) = 2(5) - 3 = 10 - 3 = 7

. Therefore, the correct answer is 7.

Step-by-Step Solution

1

Substitute

x = 1

into the definition of

g(x)

to express

g(1)

in terms of

f

.

g(1) = f(1 + 4) = f(5)

To find the value of

g(1)

, we must evaluate the input to the outer function first.

2

Substitute the input value

5

into the expression for

f(x)

.

f(5) = 2(5) - 3

Evaluating

f(5)

requires replacing

x

with

5

in the definition

f(x) = 2x - 3

.

3

Perform the operations to find the final value.

f(5) = 10 - 3 = 7

Simplifying the numerical expression gives the final value of

g(1)

.

Key Concept

Evaluating a transformed function at a given point using function notation.

Estimated Time:45s

Question 10Question →

The quadratic function $f$ is defined by $f(x) = a(x - h)^2 + k$ , where $a$ , $h$ , and $k$ are constants. In the $xy$ -plane, the graph of $y = f(x)$ has a vertex at $(3, -4)$ and passes through the point $(5, 8)$ . If the function $g$ is defined by $g(x) = -2f(x - 1) + 5$ , what is the value of $g(2)$ ?

5

7

-11

13

Show answer & explanation

Answer: -11

Answer

-11

To find the value of

g(2)

, we first determine the equation of the quadratic function

f(x)

. Since the vertex is

(3, -4)

, the vertex form is

f(x) = a(x - 3)^2 - 4

. Substituting the point

(5, 8)

yields

8 = a(5 - 3)^2 - 4

, which simplifies to

12 = 4a

, so

a = 3

. Therefore,

f(x) = 3(x - 3)^2 - 4

. We then substitute

x = 2

into the definition of

g(x)

, obtaining

g(2) = -2f(2 - 1) + 5 = -2f(1) + 5

. Evaluating

f(1)

gives

f(1) = 3(1 - 3)^2 - 4 = 8

. Substituting this back into the expression for

g(2)

gives

-2(8) + 5 = -11

. Thus, the option with value -11 is correct.

Step-by-Step Solution

1

Write the quadratic function

f(x)

in vertex form using the given vertex

(3, -4)

.

f(x) = a(x - 3)^2 - 4

The vertex form of a quadratic function is

f(x) = a(x - h)^2 + k

, where

(h, k)

is the vertex.

2

Substitute the point

(5, 8)

into the vertex form to solve for the constant

a

.

8 = a(5 - 3)^2 - 4 \implies 8 = 4a - 4 \implies 12 = 4a \implies a = 3

. Thus,

f(x) = 3(x - 3)^2 - 4

.

Since the graph of

f

passes through

(5, 8)

, these coordinates must satisfy the function's equation.

3

Substitute

x = 2

into the definition of

g(x)

to express

g(2)

in terms of

f

.

g(2) = -2f(2 - 1) + 5 = -2f(1) + 5

We need to evaluate the inner function transformation

f(x - 1)

at

x = 2

.

4

Evaluate

f(1)

using the formula determined in Step 2.

f(1) = 3(1 - 3)^2 - 4 = 3(-2)^2 - 4 = 3(4) - 4 = 8

To find

g(2)

, we must first compute the value of

f(1)

.

5

Substitute

f(1) = 8

back into the expression for

g(2)

and simplify.

g(2) = -2(8) + 5 = -16 + 5 = -11

This completes the evaluation of the multi-step transformation.

Key Concept

Function Notation and Transformations

Estimated Time:2m 0s

Question 11Question →

In the $xy$ -plane, the graph of the function $g$ is obtained by applying a sequence of transformations to the graph of the function $f(x) = |x + 2| - 3$ . Specifically, the graph of $g$ is a vertical stretch and translation of the graph of $f$ , such that $g(x) = a f(x - h) + k$ for some constants $a$ , $h$ , and $k$ . The vertex of the graph of $g$ is located at $(1, 5)$ , and the graph of $g$ passes through the point $(0, -1)$ . What is the value of $g(3)$ ?

-5

-7

17

-43

Show answer & explanation

Answer: -7

Answer

The value of

g(3)

is

-7

.

The correct value of

-7

is obtained by first identifying the vertex of

f(x) = |x + 2| - 3

at

(-2, -3)

. Comparing this to the vertex of

g(x)

at

(1, 5)

yields the horizontal shift parameter

h = 3

and the equation

-3a + k = 5

. Using the point

(0, -1)

yields the second equation

-2a + k = -1

. Solving this system gives

a = -6

and

k = -13

. Substituting these into the formula for

g(3)

yields

-7

.

Step-by-Step Solution

1

Identify the vertex of the function

f(x) = |x + 2| - 3

.

The vertex of

f(x)

is at

(-2, -3)

.

The vertex of an absolute value function of the form

y = |x - x_0| + y_0

is located at

(x_0, y_0)

.

2

Relate the vertex of

f(x)

to the vertex of

g(x) = a f(x - h) + k

.

h = 3

and

-3a + k = 5

.

The horizontal shift

h

moves the vertex from

x = -2

to

x = 1

, so

-2 + h = 1 \implies h = 3

. The vertical stretch and translation transform the

y

-coordinate of the vertex from

-3

to

5

, so

a(-3) + k = 5

.

3

Use the given point

(0, -1)

to set up a second equation.

-2a + k = -1

.

Since the graph of

g(x)

passes through

(0, -1)

, we evaluate

g(0) = a f(0 - 3) + k = -1

. Evaluating

f(-3) = |-3 + 2| - 3 = -2

yields the equation

-2a + k = -1

.

4

Solve the system of equations for

a

and

k

.

a = -6

and

k = -13

.

Subtracting

-3a + k = 5

from

-2a + k = -1

gives

a = -6

. Substituting

a = -6

back into either equation yields

k = -13

.

5

Evaluate

g(3)

using the completed function formula

g(x) = -6 f(x - 3) - 13

.

g(3) = -7

.

We substitute

x = 3

into the equation to get

g(3) = -6 f(0) - 13

. Evaluating

f(0) = |0 + 2| - 3 = -1

gives

g(3) = -6(-1) - 13 = 6 - 13 = -7

.

Key Concept

Function transformations including horizontal translations, vertical translations, and vertical scaling.

Question 12Question →

A function $f$ has exactly two local extrema: a local maximum at the point $(-2, 5)$ and a local minimum at the point $(2, -3)$ . A second function $g$ is defined by $g(x) = 1 - 3f(2x - 4)$ . If the local minimum of the graph of $y = g(x)$ occurs at the point $(h, k)$ , what is the value of $h + k$ ?

-17

-15

-13

-5

Show answer & explanation

Answer: -13

Answer

-13

To locate the local minimum of the transformed function

g(x) = 1 - 3f(2x - 4)

, we analyze the vertical reflection and scaling. The negative coefficient in

-3f(2x - 4)

reflects the graph vertically, which means the local maximum of

f(x)

at the point

(-2, 5)

becomes the local minimum of

g(x)

. To find the corresponding

x

-coordinate

h

, we solve

2h - 4 = -2

, which gives

h = 1

. To find the corresponding

y

-coordinate

k

, we evaluate

g(1) = 1 - 3f(-2) = 1 - 3(5) = -14

. Thus, the local minimum occurs at

(1, -14)

, and the sum of these coordinates is

1 + (-14) = -13

.

Step-by-Step Solution

1

Analyze how the vertical reflection in

g(x) = 1 - 3f(2x - 4)

affects the extrema.

Due to the negative coefficient in

-3f(2x - 4)

, the graph is reflected vertically. Therefore, the local maximum of

f(x)

at

(-2, 5)

transforms into the local minimum of

g(x)

, while the local minimum of

f(x)

transforms into the local maximum of

g(x)

.

A vertical reflection inverts the relative heights of the outputs, converting peaks to valleys and vice versa.

2

Determine the horizontal transformation to find the

x

-coordinate

h

of the new local minimum.

Set the input of

f

in the definition of

g(x)

equal to the

x

-coordinate of the maximum of

f(x)

, which is

-2

:

2h - 4 = -2

. Solving this equation gives

2h = 2

, which yields

h = 1

.

The horizontal shift and compression require solving for the new input variable that produces the same argument for the inner function.

3

Determine the vertical transformation to find the

y

-coordinate

k

of the new local minimum.

Substitute

h = 1

into

g(x)

to find the output value:

k = g(1) = 1 - 3f(2(1) - 4) = 1 - 3f(-2)

. Since the maximum value of

f(x)

is

f(-2) = 5

, we compute

k = 1 - 3(5) = 1 - 15 = -14

.

The vertical transformations (stretch, reflection, and shift) are applied directly to the function output.

4

Calculate the sum of the coordinates

h + k

.

Compute

h + k = 1 + (-14) = -13

.

The question asks for the sum of the coordinates of the local minimum of

g(x)

.

Key Concept

Analyzing function transformations including horizontal compression, horizontal translation, vertical stretch, reflection, and vertical translation to determine the coordinates of key features (local extrema) of a transformed function.

Estimated Time:3m 0s

Question 13Question →

A quadratic function $f$ has its vertex at $(4, -3)$ in the coordinate plane. The point $(2, 5)$ is on the graph of $y = f(x)$ . The function $g$ is defined by $g(x) = -f(x - 3) + 8$ . What is the value of $g(8)$ ?

Show answer & explanation

Answer: 9

Answer

The correct answer is 9.

The correct answer is 9. To find this, we first establish the vertex form of the quadratic function

f(x) = a(x - 4)^2 - 3

. Substituting the point

(2, 5)

gives

5 = a(2 - 4)^2 - 3

, which simplifies to

8 = 4a

, so

a = 2

. Thus,

f(x) = 2(x - 4)^2 - 3

. To find the value of

g(8)

, we substitute

8

into the definition of

g(x)

, yielding

g(8) = -f(8 - 3) + 8 = -f(5) + 8

. Evaluating

f(5)

gives

2(5 - 4)^2 - 3 = -1

. Finally, substituting

-1

back into the expression for

g(8)

gives

-(-1) + 8 = 9

.

Step-by-Step Solution

1

Write the quadratic function

f(x)

in vertex form using its vertex

(4, -3)

.

f(x) = a(x - 4)^2 - 3

The vertex form of a quadratic function with vertex

(h, k)

is

f(x) = a(x - h)^2 + k

.

2

Substitute the coordinates of the point

(2, 5)

into the vertex form to find the value of the constant

a

.

a = 2

, so

f(x) = 2(x - 4)^2 - 3

Since the point

(2, 5)

lies on the graph of

f

, substituting

x = 2

and

f(2) = 5

allows us to solve for the vertical stretch factor

a

.

3

Use the definition of

g(x)

to express

g(8)

in terms of

f

.

g(8) = -f(5) + 8

Substituting

x = 8

into the equation

g(x) = -f(x - 3) + 8

yields

g(8) = -f(8 - 3) + 8 = -f(5) + 8

.

4

Evaluate

f(5)

using the equation for

f(x)

found in Step 2.

f(5) = -1

Substituting

x = 5

into

f(x) = 2(x - 4)^2 - 3

gives

2(5 - 4)^2 - 3 = 2(1) - 3 = -1

.

5

Substitute the value of

f(5)

into the expression for

g(8)

to find the final result.

g(8) = 9

Substituting

f(5) = -1

into

g(8) = -f(5) + 8

yields

-(-1) + 8 = 1 + 8 = 9

.

Key Concept

Finding the equation of a quadratic function from its vertex and a point, and evaluating transformed functions using function notation.

Question 14Question →

Several values of $x$ and the corresponding values of $f(x)$ for the quadratic function $f$ are shown in the table below.

$x$	$f(x)$
$-4$	$18$
$-2$	$6$
$0$	$2$
$2$	$6$
$4$	$18$

The function $g$ is defined by $g(x) = -2(f(x - 3) + 4)$ . If the vertex of the graph of $y = f(x)$ corresponds to the point $(p, q)$ on the graph of $y = g(x)$ , what is the value of $p + q$ ?

-15

-9

-5

3

Show answer & explanation

Answer: $-9$

Answer

-9

The vertex of the quadratic function

f(x)

is identified from the symmetric table values as

(0, 2)

. Under the transformation

g(x) = -2(f(x - 3) + 4)

, the input

x - 3

shifts the vertex horizontally to the right by 3 units, so the new x-coordinate is

p = 3

. Evaluating

g(3)

gives

q = -2(f(0) + 4) = -2(2 + 4) = -2(6) = -12

. The sum of these coordinates is

p + q = 3 + (-12) = -9

.

Step-by-Step Solution

1

Identify the vertex of the function

f(x)

from the table.

Vertex of

f(x)

is

(0, 2)

.

Since

f

is a quadratic function and the table shows symmetry about

x = 0

(with

f(-2) = f(2) = 6

and

f(-4) = f(4) = 18

), the vertex must be at the point where

x = 0

, which gives

f(0) = 2

.

2

Determine the x-coordinate

p

of the corresponding point on the graph of

g(x)

.

p = 3

.

The function

g

is defined as

g(x) = -2(f(x - 3) + 4)

. The expression

f(x - 3)

indicates a horizontal translation of the graph of

f

to the right by 3 units. Therefore, the x-coordinate of the vertex shifts from

0

to

0 + 3 = 3

.

3

Determine the y-coordinate

q

of the corresponding point on the graph of

g(x)

by evaluating

g(3)

.

q = -12

.

Substitute

x = 3

into the definition of

g(x)

:

g(3) = -2(f(3 - 3) + 4) = -2(f(0) + 4)

. Since

f(0) = 2

, this simplifies to

-2(2 + 4) = -2(6) = -12

.

4

Calculate the value of

p + q

.

p + q = -9

.

Adding the coordinates

p = 3

and

q = -12

yields

3 + (-12) = -9

.

Key Concept

Function Notation and Transformations

Question 15Question →

The function $f$ is defined by $f(x) = 3^x - 4$ . In the $xy$ -plane, the graph of the function $g$ is obtained by first reflecting the graph of $f$ across the $x$ -axis, then translating the graph vertically up by 10 units, and finally translating the graph horizontally to the right by 2 units. If $g(c) = 5$ , what is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

The value of c is 4.

Reflecting the function

f(x) = 3^x - 4

across the

x

-axis changes its sign to

-f(x) = -3^x + 4

. Translating this graph vertically up by 10 units adds 10 to the function, yielding

-3^x + 14

. Finally, translating horizontally to the right by 2 units replaces

x

with

x - 2

, producing the function

g(x) = -3^{x-2} + 14

. Setting

g(c) = 5

gives the equation

-3^{c-2} + 14 = 5

. Subtracting 14 from both sides results in

-3^{c-2} = -9

, which simplifies to

3^{c-2} = 9

. Since

9 = 3^2

, the exponent

c-2

must equal 2, which gives

c = 4

.

Step-by-Step Solution

1

Reflect the function

f(x) = 3^x - 4

across the

x

-axis

-f(x) = -(3^x - 4) = -3^x + 4

A reflection across the

x

-axis replaces

y

with

-y

, meaning the entire function is multiplied by

-1

.

2

Translate the reflected function vertically up by 10 units

-3^x + 4 + 10 = -3^x + 14

A vertical translation upward by

k

units adds

k

directly to the function expression.

3

Translate the function horizontally to the right by 2 units

g(x) = -3^{x-2} + 14

A horizontal translation to the right by

h

units replaces

x

with

x-h

in the function expression.

4

Set

g(c) = 5

and solve the exponential equation for

c

-3^{c-2} + 14 = 5 \implies 3^{c-2} = 9 \implies c - 2 = 2 \implies c = 4

Substitute

c

into

g(x)

, set the output to 5, isolate the exponential term, and equate exponents to find

c

.

Key Concept

Applying sequential function transformations (reflections, vertical translations, horizontal translations) algebraically and solving exponential equations.

Question 16Question →

The graph of $y = f(x)$ has a relative minimum at the point $(2, -4)$ in the $xy$ -plane. If the function $g$ is defined by $g(x) = 3 - f(2x - 6)$ , what are the coordinates of the corresponding relative maximum on the graph of $y = g(x)$ ?

(4, 7)

(-1, 7)

(4, -1)

(6, 7)

Show answer & explanation

Answer: $(4, 7)$

Answer

(4, 7)

To find the coordinates of the relative maximum on the graph of

y = g(x)

that corresponds to the relative minimum of

y = f(x)

at

(2, -4)

, we first find the new

x

-coordinate. The input to the function

f

must be

2

, so we set the argument

2x - 6 = 2

, which solves to

x = 4

. Next, we find the new

y

-coordinate by evaluating

g(4) = 3 - f(2)

. Substituting

f(2) = -4

, we get

g(4) = 3 - (-4) = 7

. Thus, the coordinates of the corresponding relative maximum are

(4, 7)

.

Step-by-Step Solution

1

Set the argument of the function

f

in

g(x)

equal to the

x

-coordinate of the known point on the graph of

f

.

2x - 6 = 2

The relative minimum of

f

occurs when its input is 2.

2

Solve the equation for

x

to find the transformed

x

-coordinate.

2x = 8 \implies x = 4

Isolating the variable

x

by adding 6 to both sides and then dividing by 2.

3

Substitute

x = 4

into the definition of

g(x)

to calculate the transformed

y

-coordinate.

g(4) = 3 - f(2(4) - 6) = 3 - f(2)

This evaluates the vertical transformation of the function at the corresponding

x

-value.

4

Substitute the value

f(2) = -4

and simplify.

g(4) = 3 - (-4) = 3 + 4 = 7

Since the minimum of

f

is at

(2, -4)

, we know

f(2) = -4

. The negation in front of

f

reflects the minimum to a maximum.

Key Concept

Function Notation and Transformations

Estimated Time:2m 0s

Question 17Question →

The graph of the function $f$ in the $xy$ -plane has a vertex at $(-2, 7)$ . The function $g$ is defined by $g(x) = f(-x + 3) + 12$ . If the vertex of the graph of $y = g(x)$ is the point $(a, b)$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 24

Answer

24

The vertex of the parent function

f(x)

is at

(-2, 7)

, which means

f(-2) = 7

. The transformed function is

g(x) = f(-x + 3) + 12

. The vertex of

g(x)

occurs when the input to

f

, which is

-x + 3

, is equal to

-2

. Solving

-x + 3 = -2

gives

-x = -5

, or

x = 5

, so the x-coordinate of the vertex of

g(x)

is

a = 5

. To find the y-coordinate

b

, we evaluate

g(5) = f(-(5) + 3) + 12 = f(-2) + 12 = 7 + 12 = 19

. Therefore, the vertex of the graph of

y = g(x)

is

(5, 19)

, so

a = 5

and

b = 19

. The value of

a + b

is

5 + 19 = 24

.

Step-by-Step Solution

1

Find the x-coordinate of the vertex of the transformed function

g(x)

by setting the inner expression equal to the x-coordinate of the original vertex.

a = 5

The vertex of

f(x)

is located at

x = -2

. For

g(x) = f(-x + 3) + 12

, the vertex occurs when the input to

f

,

-x + 3

, is equal to

-2

. Solving

-x + 3 = -2

yields

x = 5

.

2

Find the y-coordinate of the vertex of

g(x)

by evaluating

g(5)

.

b = 19

Substituting

x = 5

into the definition of

g(x)

gives

g(5) = f(-2) + 12

. Since the vertex of

f

is at

(-2, 7)

,

f(-2) = 7

. Thus,

g(5) = 7 + 12 = 19

.

3

Calculate the sum of the coordinates

a

and

b

.

24

The vertex of

g(x)

is

(5, 19)

, so

a = 5

and

b = 19

. The sum

a + b

is

5 + 19 = 24

.

Key Concept

Determining the coordinates of a transformed vertex using function notation.

Question 18Question →

For the function $f$ , selected values of $x$ and $f(x)$ are shown in the table below.

$x$	$f(x)$
$-1$	$4$
$1$	$2$
$3$	$-1$
$5$	$6$

The function $g$ is defined by $g(x) = a \cdot f(x - 2) + 5$ , where $a$ is a constant. If $g(5) = 3$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

2

Evaluating

g(5)

using the formula

g(x) = a \cdot f(x - 2) + 5

requires finding

f(5 - 2) = f(3)

. From the table,

f(3) = -1

. Substituting these values yields

3 = a(-1) + 5

, which simplifies to

-a = -2

, or

a = 2

.

Step-by-Step Solution

1

Express

g(5)

using the given definition of

g(x)

.

g(5) = a \cdot f(3) + 5

By substituting

x = 5

into the definition

g(x) = a \cdot f(x - 2) + 5

, we obtain

g(5) = a \cdot f(5 - 2) + 5 = a \cdot f(3) + 5

.

2

Find the value of

f(3)

from the table.

f(3) = -1

Looking at the row where

x = 3

in the table, the corresponding output

f(x)

is

-1

.

3

Substitute the known values into the equation for

g(5)

and solve for

a

.

a = 2

Substitute

g(5) = 3

and

f(3) = -1

into the equation to get

3 = a(-1) + 5

. Subtracting 5 from both sides gives

-2 = -a

, which simplifies to

a = 2

.

Key Concept

Evaluating a transformed function using a table of values.

Question 19Question →

The function $f$ is defined by $f(x) = x^2 - 4x + 7$ . The function $g$ is defined by $g(x) = f(x - 3) + 2$ . If the minimum value of $f$ is $v$ , and the minimum value of $g$ occurs at $x = k$ , what is the value of $v + k$ ?

1

2

7

8

Show answer & explanation

Answer: 8

Answer

8

To find the minimum value of

f(x) = x^2 - 4x + 7

, we can complete the square to write it in vertex form:

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

. The vertex of the graph of

f

is

(2, 3)

, meaning the minimum value of

f

is

v = 3

, occurring at

x = 2

. The function

g

is defined as

g(x) = f(x - 3) + 2

, which represents a shift of the graph of

f

to the right by 3 units and up by 2 units. Since the minimum of

f

occurs at

x = 2

, the minimum of

g

occurs at

x = 2 + 3 = 5

, so

k = 5

. Thus, the value of

v + k

is

3 + 5 = 8

.

Step-by-Step Solution

1

Find the vertex form of the quadratic function

f(x)

to identify its minimum value

v

and the x-coordinate where it occurs.

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

, so the minimum value is

v = 3

, which occurs at

x = 2

.

Completing the square allows us to read the vertex

(h, k)

of the parabola directly, where

h

is the x-coordinate of the vertex and

k

is the minimum value.

2

Determine the x-coordinate

k

where the minimum value of

g(x)

occurs using function transformations.

k = 2 + 3 = 5

The function

g(x) = f(x - 3) + 2

shifts the graph of

f

to the right by 3 units. Therefore, the minimum point is translated from

x = 2

to

x = 2 + 3 = 5

.

3

Calculate the sum

v + k

.

3 + 5 = 8

Adding the minimum value of

f

(

v = 3

) to the x-coordinate of the minimum of

g

(

k = 5

) gives the final required value.

Key Concept

Identifying the vertex and minimum values of quadratic functions, and applying horizontal and vertical translations to their graphs.

Estimated Time:1m 30s

Question 20Question →

Let the function $f$ be given by $f(x) = 3x - 5$ . If a second function $g$ is defined in terms of $f$ as $g(x) = 2f(x + 1) + 4$ , what is the value of $x$ for which $g(x) = 12$ ?

Show answer & explanation

Answer: 2

Answer

2

Evaluating

g(x) = 12

gives

2f(x + 1) + 4 = 12

. Subtracting 4 from both sides yields

2f(x + 1) = 8

. Dividing by 2 yields

f(x + 1) = 4

. Substituting

x+1

into the definition

f(x) = 3x - 5

gives

3(x + 1) - 5 = 4

. Simplifying the equation results in

3x + 3 - 5 = 4

, which is

3x - 2 = 4

. Solving for

x

gives

3x = 6

, so

x = 2

.

Step-by-Step Solution

1

Set the equation

g(x) = 12

using the definition of

g(x)

2f(x + 1) + 4 = 12

We are given that

g(x) = 12

and want to find the corresponding value of

x

.

2

Isolate the function term

f(x + 1)

f(x + 1) = 4

Subtract 4 from both sides of the equation to get

2f(x + 1) = 8

, then divide by 2.

3

Use the definition of

f(x)

to express

f(x + 1)

f(x + 1) = 3(x + 1) - 5 = 3x - 2

Substitute

x + 1

in place of

x

in the function

f(x) = 3x - 5

.

4

Set the expression for

f(x + 1)

equal to 4 and solve for

x

x = 2

Solve the linear equation

3x - 2 = 4

by adding 2 to both sides to get

3x = 6

, then dividing by 3.

Key Concept

Applying multiple transformations to a linear function and solving the resulting equation using function notation.

Function Notation and Transformations

Answer

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