The table below shows some values for a linear function f, where k is a positive constant. | x | f(x) | |---|---| | 0 | 3 | | k | 2k + 6 | | 4k | 27 | What is the slope of the graph of y = f(x) in the xy-plane?

The table below shows some values for a linear function f, where k is a positive constant.… — SAT

The table below shows some values for a linear function $f$ , where $k$ is a positive constant.

$x$	$f(x)$
$0$	$3$
$k$	$2k + 6$
$4k$	$27$

What is the slope of the graph of $y = f(x)$ in the $xy$ -plane?

A
2
B
-4
4Answer
D
$\frac{1}{4}$

Answer

The slope of the graph of

f

4

The correct answer is

4

. A linear function can be written in the form

f(x) = mx + b

, where

m

is the slope and

b

is the

y

-intercept. Since the

y

-intercept of the graph of

f

(0, 3)

, we have

b = 3

, so the function is

f(x) = mx + 3

. Using the given points, we can set up the equations

f(k) = mk + 3 = 2k + 6

and

f(4k) = 4mk + 3 = 27

. Solving this system yields

mk = 6

and

2k = 3

, which gives

k = 1.5

. Substituting

k = 1.5

into

mk = 6

gives the slope

m = 4

Step-by-Step Solution

Express the linear function

f(x)

using the slope-intercept form and the given

y

-intercept.

f(x) = mx + 3

, where

m

is the slope and the

y

-intercept is

(0, 3)

A linear function has the general form

f(x) = mx + b

. The table indicates that when

x = 0

f(x) = 3

, which gives the

y

-intercept

(0, 3)

and establishes

b = 3

Use the point

(k, 2k + 6)

from the table to write an equation involving

m

and

k

mk + 3 = 2k + 6 \implies mk = 2k + 3

Substituting

x = k

into the function expression gives

f(k) = mk + 3

. Setting this equal to the table value

2k + 6

allows us to express

mk

in terms of

k

Use the point

(4k, 27)

from the table to write another equation involving

m

and

k

, and solve for the product

mk

4mk + 3 = 27 \implies 4mk = 24 \implies mk = 6

Substituting

x = 4k

into the function expression gives

f(4k) = 4mk + 3

. Setting this equal to the table value

27

allows us to solve directly for the numerical value of

mk

Substitute the value of

mk

into the equation from Step 2 to solve for the constant

k

6 = 2k + 3 \implies 2k = 3 \implies k = 1.5

By replacing

mk

with

6

in the equation

mk = 2k + 3

, we get a single-variable linear equation that we can solve for

k

Solve for the slope

m

using the value of

k

and the product

mk

m(1.5) = 6 \implies m = 4

Since

mk = 6

and

k = 1.5

, dividing the product

6

1.5

yields the slope

m

Key Concept

Using coordinate points and intercepts to determine the slope of a linear function represented in a table.

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Key Concept

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