The graph of the linear function f in the xy-plane is defined by f(x) = kx + 4, where k is… — SAT

The graph of the linear function $f$ in the $xy$ -plane is defined by $f(x) = kx + 4$ , where $k$ is a positive constant. The graph of the linear function $g$ is obtained by translating the graph of $f$ left by $3$ units and down by $2$ units. If the $x$ -intercept of the graph of $g$ is $5$ times the $x$ -intercept of the graph of $f$ , what is the value of $k$ ?

Answer: 6

Answer

The correct answer is 6. The linear function

g(x)

represents the translated graph, which is expressed as

g(x) = f(x+3) - 2

. Substituting

f(x) = kx + 4

yields

g(x) = k(x+3) + 4 - 2 = kx + 3k + 2

. The

x

-intercept of a graph is the point where the function value is

0

. Solving

f(x) = 0

gives the

x

-intercept of the graph of

f

x = -\frac{4}{k}

. Solving

g(x) = 0

gives the

x

-intercept of the graph of

g

x = -\frac{3k + 2}{k}

. We are given that the

x

-intercept of the graph of

g

5

times the

x

-intercept of the graph of

f

, which translates to the equation

-\frac{3k+2}{k} = 5 \left(-\frac{4}{k}\right)

. Multiplying both sides by

-k

(since

k > 0

k \neq 0

) yields

3k + 2 = 20

. Subtracting

2

from both sides gives

3k = 18

, and dividing by

3

results in

k = 6

Step-by-Step Solution

Determine the equation of the translated function

g(x)

g(x) = kx + 3k + 2

Shifting the graph of

f(x)

left by

3

units and down by

2

units translates to

g(x) = f(x+3) - 2

Find the

x

-intercept of the graph of

f

x = -\frac{4}{k}

The

x

-intercept is the input value for which the output of the function is zero, so we solve

f(x) = 0

Find the

x

-intercept of the graph of

g

x = -\frac{3k + 2}{k}

We set the equation of

g(x)

to zero and solve for

x

in terms of

k

Set up and solve the equation relating the two

x

-intercepts

k = 6

Using the relation

x_g = 5 \cdot x_f

, we obtain

-\frac{3k + 2}{k} = 5 \left(-\frac{4}{k}\right)

, which simplifies to

3k + 2 = 20

, leading to

k = 6

Key Concept

Function translations and finding intercepts of linear equations

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