The tables below show some values of the function f and the transformed function g, where … — SAT

The tables below show some values of the function $f$ and the transformed function $g$ , where $g(x) = f(x - h) + k$ for constants $h$ and $k$ .

$x$	$f(x)$
$-2$	$8$
$0$	$3$
$2$	$-1$
$4$	$5$

$x$	$g(x)$
$1$	$6$
$3$	$1$
$5$	$-3$
$7$	$3$

What is the value of $h + k$ ?

A
5
1Answer
C
-5
D
-1

Answer

The correct answer is 1. By comparing the tables, each input of

g(x)

3

units greater than the corresponding input of

f(x)

(for example,

g(1)

corresponds to

f(-2)

because

1 - (-2) = 3

). A horizontal shift of

3

units to the right is represented in function notation by subtracting

3

from the input variable, so

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

, which gives

h = 3

. Next, comparing the output values shows that each output of

g(x)

2

units less than the corresponding output of

f(x)

(for example,

g(1) = 6

while

f(-2) = 8

, and

6 = 8 - 2

). This indicates a vertical shift downwards by

2

units, so

k = -2

. Therefore, the value of

h + k

3 + (-2) = 1

Step-by-Step Solution

Determine the horizontal translation constant

h

by comparing the inputs of

f(x)

and

g(x)

that correspond to related output values.

The inputs for

f(x)

are

\{-2, 0, 2, 4\}

and the corresponding inputs for

g(x)

are

\{1, 3, 5, 7\}

. Each input for

g(x)

3

units greater than the corresponding input for

f(x)

(since

1 - (-2) = 3

3 - 0 = 3

, etc.). This indicates a horizontal shift of

3

units to the right, which means the argument of

f

g(x)

must be

x - 3

. Comparing this with

f(x - h)

gives

h = 3

Identifying the horizontal shift determines the value of the parameter

h

in the translation equation

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

Determine the vertical translation constant

k

by comparing the output values of

g(x)

with the corresponding values of

f(x - 3)

Using the matching inputs, compare the outputs:
- For

x = 1

g(1) = 6

and

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

. The difference is

6 - 8 = -2

.
- For

x = 3

g(3) = 1

and

f(3 - 3) = f(0) = 3

. The difference is

1 - 3 = -2

.
- For

x = 5

g(5) = -3

and

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

. The difference is

-3 - (-1) = -2

.
- For

x = 7

g(7) = 3

and

f(7 - 3) = f(4) = 5

. The difference is

3 - 5 = -2

.
Since each output of

g(x)

2

units less than the corresponding output of

f(x - 3)

, the vertical translation is

k = -2

Comparing the corresponding outputs identifies the vertical shift constant

k

in the translation equation

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

Calculate the sum of the constants

h

and

k

h + k = 3 + (-2) = 1

Evaluating the sum of the parameters provides the final value requested by the question.

Key Concept

Function Notation and Transformations

Alternative Method

Instead of analyzing the general shift for all points, you can choose a single corresponding pair of points from the tables to set up equations. Let the point

(-2, 8)

from the table of

f

correspond to the point

(1, 6)

from the table of

g

. Since

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

, we substitute

x = 1

to get

g(1) = f(1 - h) + k

. Substituting the values

g(1) = 6

and setting the input of

f

to match, we get

1 - h = -2 \implies h = 3

. This simplifies the equation to

6 = f(-2) + k \implies 6 = 8 + k \implies k = -2

. This gives

h + k = 3 - 2 = 1

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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