The linear function f is defined such that its graph in the xy-plane is perpendicular to t… — SAT

The linear function $f$ is defined such that its graph in the $xy$ -plane is perpendicular to the line $3x - 4y = 20$ . The graph of another linear function, $g$ , is the result of shifting the graph of $f$ right by $5$ units and down by $4$ units. If $f(0) = 8$ and the graph of $g$ intersects the $x$ -axis at the point $(k, 0)$ , what is the value of $k$ ?

Answer: 8

Answer

The slope of the line

3x - 4y = 20

\frac{3}{4}

. The slope of a line perpendicular to it is the negative reciprocal,

-\frac{4}{3}

. With a

y

-intercept of

(0, 8)

, the function is

f(x) = -\frac{4}{3}x + 8

. Translating this function

5

units right and

4

units down yields

g(x) = f(x - 5) - 4 = -\frac{4}{3}(x - 5) + 4 = -\frac{4}{3}x + \frac{32}{3}

. Setting

g(k) = 0

to find the

x

-intercept gives

-\frac{4}{3}k + \frac{32}{3} = 0

, which solves to

k = 8

Step-by-Step Solution

Find the slope of the line

3x - 4y = 20

The slope is

\frac{3}{4}

Converting the line equation to slope-intercept form

y = mx + b

gives

y = \frac{3}{4}x - 5

, where the coefficient of

x

represents the slope.

Find the slope of

f

The slope of

f

-\frac{4}{3}

Perpendicular lines have slopes that are negative reciprocals of each other.

Write the function

f(x)

f(x) = -\frac{4}{3}x + 8

Since the

y

-intercept is given by

f(0) = 8

and the slope is

-\frac{4}{3}

, the slope-intercept form is

f(x) = -\frac{4}{3}x + 8

Determine the function

g(x)

by translating

f(x)

5

units right and

4

units down.

g(x) = -\frac{4}{3}x + \frac{32}{3}

A translation of

f(x)

right by

5

units and down by

4

units corresponds to

g(x) = f(x - 5) - 4

. Substituting

x-5

into

f(x)

gives

g(x) = -\frac{4}{3}(x - 5) + 8 - 4 = -\frac{4}{3}x + \frac{32}{3}

Set

g(k) = 0

and solve for

k

k = 8

The graph of

g

intersects the

x

-axis at

(k, 0)

, which means

g(k) = 0

. Solving

-\frac{4}{3}k + \frac{32}{3} = 0

yields

k = 8

Key Concept

Writing linear functions using perpendicular slopes and performing vertical and horizontal transformations.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic