The table below shows some values of the linear function h. | x | h(x) | |---|---| | 2 | k - 4 | | 5 | k + 8 | | 8 | 2k + 2 | If k is a constant, what is the value of k?

The table below shows some values of the linear function h. | x | h(x) | |---|---| | 2 | k… — SAT

The table below shows some values of the linear function $h$ .

If $k$ is a constant, what is the value of $k$ ?

Answer: 18

The value of the constant

k

18

Since the function

h

is linear, its rate of change (slope) is constant. Calculating the slope using the first two coordinate pairs

(2, k - 4)

and

(5, k + 8)

gives

\frac{(k + 8) - (k - 4)}{5 - 2} = \frac{12}{3} = 4

. Using the next two coordinate pairs

(5, k + 8)

and

(8, 2k + 2)

gives the slope as

\frac{(2k + 2) - (k + 8)}{8 - 5} = \frac{k - 6}{3}

. Setting these two slope values equal to each other gives the equation

\frac{k - 6}{3} = 4

. Multiplying both sides by

3

results in

k - 6 = 12

, and adding

6

to both sides yields

k = 18

Identify that the rate of change (slope) of a linear function is constant between any two points.

The slope calculated from the first two points must equal the slope calculated from the second and third points.

This relationship allows us to set up an algebraic equation to solve for the unknown constant

k

Calculate the slope using the first two points:

(2, k - 4)

and

(5, k + 8)

Slope =

\frac{(k + 8) - (k - 4)}{5 - 2} = \frac{k + 8 - k + 4}{3} = \frac{12}{3} = 4

This simplifies to a constant numerical value of

4

for the slope of the function.

Calculate the slope using the second and third points:

(5, k + 8)

and

(8, 2k + 2)

Slope =

\frac{(2k + 2) - (k + 8)}{8 - 5} = \frac{2k + 2 - k - 8}{3} = \frac{k - 6}{3}

This provides a second expression for the slope in terms of the variable

k

Equate the two slope expressions and solve for

k

\frac{k - 6}{3} = 4 \implies k - 6 = 12 \implies k = 18

Setting the two expressions equal and solving isolating

k

gives the correct value of

18

A linear function has a constant rate of change (slope) between any two points on its graph.