The table below shows some values of x and the corresponding values of f(x) for a linear f… — SAT

The table below shows some values of $x$ and the corresponding values of $f(x)$ for a linear function $f$ .

What is the value of $f(15)$ ?

Answer: 27

The value of the linear function evaluated at 15 is 27.

To find

f(15)

for the linear function, we determine its constant rate of change. Using points

(3, 7)

and

(6, 12)

, the change in

f(x)

12 - 7 = 5

for a change in

x

6 - 3 = 3

. This gives a slope of

\frac{5}{3}

. Using point-slope form with

(3, 7)

gives the equation

f(x) - 7 = \frac{5}{3}(x - 3)

, which simplifies to

f(x) = \frac{5}{3}x + 2

. Substituting

x = 15

yields

f(15) = \frac{5}{3}(15) + 2 = 25 + 2 = 27

. Alternatively, since

x

increases by 6 from 9 to 15, which is twice the step size of 3, the value of

f(x)

must increase by

2 \times 5 = 10

from

17

, resulting in

17 + 10 = 27

Calculate the slope of the linear function

slope

m = \frac{5}{3}

Linear functions have a constant rate of change, which is the slope.

Determine the equation of the function

f(x) = \frac{5}{3}x + 2

Using the slope and one point from the table helps define the function for all inputs.

Evaluate the function at

x = 15

f(15) = 27

Substitute the given input value into the function equation to find the corresponding output value.

Linear Functions and Graphs

Question