Question

consider the graph of f(x) given below.
the function g(x) is a transformation of f(x). if g(x) has a y-intercept at 3, which of the following functions could represent g(x)?
a. \\(g(x) = f(x) + 3\\)
b. \\(g(x) = f(x) + 4\\)
c. \\(g(x) = f(x - 4)\\)
d. \\(g(x) = f(x - 3)\\)

Explanation:

Step1: Find $f(x)$ equation

From the graph, the line has slope $m=1$ (rises 1, runs 1) and y-intercept $b=-1$. So $f(x)=x-1$.

Step2: Find y-intercept of $f(x)$

The y-intercept of $f(x)$ is $f(0)=0-1=-1$.

Step3: Test vertical shift options

For vertical shifts $g(x)=f(x)+k$, the new y-intercept is $-1+k$. Set equal to 3:
$-1+k=3 \implies k=4$. So $g(x)=f(x)+4$.

Step4: Test horizontal shift options

For $g(x)=f(x-a)$, y-intercept is $f(-a)=-a-1$. Set to 3:
$-a-1=3 \implies a=-4$, which does not match options C/D.

Answer:

B. $g(x) = f(x) + 4$