The table below shows some values for the linear function f. | x | f(x) | | :--- | :--- | … — SAT

The table below shows some values for the linear function $f$ .

$x$	$f(x)$
$2$	$5$
$5$	$14$
$8$	$23$

If the function $g$ is defined by $g(x) = f(x) + k$ , where $k$ is a constant, and the $x$ -intercept of the graph of $y = g(x)$ in the $xy$ -plane is $-2$ , what is the value of $k$ ?

Answer: 7

Answer

The value of

k

7

To find the value of

k

, we first determine the linear function

f(x)

using the values in the table. The slope of

f

is calculated as

\frac{14 - 5}{5 - 2} = 3

. Substituting the point

(2, 5)

into the slope-intercept form

f(x) = 3x + b

gives

5 = 3(2) + b

, which simplifies to

b = -1

. Therefore,

f(x) = 3x - 1

. The function

g

is defined as

g(x) = f(x) + k = 3x - 1 + k

. Since the

x

-intercept of the graph of

y = g(x)

-2

, we must have

g(-2) = 0

. Substituting

x = -2

into the equation for

g(x)

gives

3(-2) - 1 + k = 0

, which simplifies to

-7 + k = 0

. Solving for

k

yields

k = 7

Step-by-Step Solution

Determine the slope of the linear function

f

The slope of

f

3

Using the points

(2, 5)

and

(5, 14)

from the table, the slope is calculated as

m = \frac{14 - 5}{5 - 2} = \frac{9}{3} = 3

Find the equation for

f(x)

f(x) = 3x - 1

Using the slope-intercept form

f(x) = 3x + b

and the point

(2, 5)

, we solve for

b

5 = 3(2) + b

, which gives

b = -1

Set up the equation for

g(x)

and use the

x

-intercept to find the value of

k

k = 7

Since

g(x) = f(x) + k

, we have

g(x) = 3x - 1 + k

. The

x

-intercept of

g(x)

-2

, so

g(-2) = 0

. This gives

3(-2) - 1 + k = 0

, which simplifies to

-7 + k = 0

, meaning

k = 7

Key Concept

Finding the equation of a linear function from tabular data and applying translations and intercepts.

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