A linear function f is defined such that its graph passes through the points (2, 5) and (4… — SAT

A linear function $f$ is defined such that its graph passes through the points $(2, 5)$ and $(4, 11)$ . The graph of another linear function, $g$ , is perpendicular to the graph of $f$ . If the graph of $g$ passes through the point $(3, 10)$ , what is the $x$ -coordinate of the $x$ -intercept of the graph of $g$ ?

Answer: 33

Answer

The correct answer is

33

The slope of the linear function

f

is found using the two given points:

\frac{11 - 5}{4 - 2} = 3

. Because the graph of the linear function

g

is perpendicular to the graph of

f

, its slope is the negative reciprocal of

3

, which is

-\frac{1}{3}

. Using the point-slope form with the point

(3, 10)

, the equation of

g

y - 10 = -\frac{1}{3}(x - 3)

, which simplifies to

y = -\frac{1}{3}x + 11

. The

x

-intercept is the value of

x

when

y = 0

. Setting

y

0

gives

0 = -\frac{1}{3}x + 11

, which yields

x = 33

Step-by-Step Solution

Calculate the slope of the linear function

f

The slope of

f

3

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

\frac{y_2 - y_1}{x_2 - x_1}

. Substituting

(2, 5)

and

(4, 11)

yields

\frac{11 - 5}{4 - 2} = \frac{6}{2} = 3

Determine the slope of the perpendicular linear function

g

The slope of

g

-\frac{1}{3}

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other. The negative reciprocal of

3

-\frac{1}{3}

Write the equation representing function

g

y = -\frac{1}{3}x + 11

Using the point-slope equation

y - y_1 = m(x - x_1)

with the slope

m = -\frac{1}{3}

and the point

(3, 10)

, we get

y - 10 = -\frac{1}{3}(x - 3)

. Distributing and simplifying yields

y - 10 = -\frac{1}{3}x + 1 \implies y = -\frac{1}{3}x + 11

Find the

x

-intercept of the graph of

g

33

The

x

-intercept is the point on the graph where

y = 0

. Substituting

0

for

y

in the equation of

g

gives

0 = -\frac{1}{3}x + 11 \implies \frac{1}{3}x = 11 \implies x = 33

Key Concept

The relationship between the slopes of perpendicular lines and finding intercepts of linear functions.

Estimated Time:1m 30s

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