In the xy-plane, the graph of a linear function f passes through the point (1, 3) and has … — SAT

In the $xy$ -plane, the graph of a linear function $f$ passes through the point $(1, 3)$ and has a slope of $m$ , where $m \neq 0$ . The graph of a second linear function, $g$ , is obtained by reflecting the graph of $f$ across the $y$ -axis and then translating it up by $4$ units. If the graphs of $f$ and $g$ intersect at a point on the $x$ -axis, what is the value of $m$ ?

A
1
B
-2
5Answer
D
3

Answer

The correct value is

5

. By using the point-slope form, the first function is

f(x) = mx - m + 3

. Reflecting it across the

y

-axis gives

f(-x) = -mx - m + 3

, and translating it up by

4

units yields

g(x) = -mx - m + 7

. Since they intersect on the

x

-axis, the

y

-value at their intersection point is

0

. This gives the system of equations

mx_0 = m - 3

and

-mx_0 = m - 7

. Adding these equations eliminates the

mx_0

term, leaving

2m - 10 = 0

, which simplifies to

m = 5

Step-by-Step Solution

Write the equation of the linear function

f(x)

using the point-slope form.

f(x) = m(x - 1) + 3 = mx - m + 3

The function passes through the point

(1, 3)

and has a slope of

m

Determine the equation of the linear function

g(x)

by applying the reflection and translation transformations.

g(x) = f(-x) + 4 = -mx - m + 7

Reflecting across the

y

-axis replaces

x

with

-x

, and translating up by

4

units adds

4

to the function.

Set the

y

-coordinates of the intersection point

(x_0, 0)

on the

x

-axis to

0

for both functions.

mx_0 - m + 3 = 0 \implies mx_0 = m - 3

and

-mx_0 - m + 7 = 0 \implies -mx_0 = m - 7

An intersection on the

x

-axis means that the

y

-coordinate of the intersection point is

0

Solve the system of equations to find the value of

m

0 = (m - 3) + (m - 7) \implies 2m = 10 \implies m = 5

Adding the two equations eliminates the term

mx_0

, allowing us to solve directly for

m

Key Concept

Applying transformations (reflections and translations) to linear functions and finding their intercepts in the coordinate plane.

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