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 - 3)
d. g(x) = f(x) + 4

Explanation:

Step1: Find y-intercept of \( f(x) \)

From the graph, when \( x = 0 \), \( f(0)= - 1 \) (since the line crosses the y - axis at \( (0,-1) \)).

Step2: Analyze each option for the y - intercept of \( g(x) \)

• Option A: \( g(x)=f(x) + 3 \). The y - intercept of \( g(x) \) is \( f(0)+3=-1 + 3=2

eq3 \).

• Option B: \( g(x)=f(x - 4) \). To find the y - intercept, set \( x = 0 \), so \( g(0)=f(-4) \). From the graph of \( f(x) \) (a linear function, let's find its equation. The slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \), using points \( (0,-1) \) and \( (2,1) \), \( m=\frac{1-(-1)}{2 - 0}=\frac{2}{2}=1 \). So \( f(x)=x - 1 \). Then \( f(-4)=-4-1=-5 \), so \( g(0)=-5

eq3 \).

• Option C: \( g(x)=f(x - 3) \). Set \( x = 0 \), \( g(0)=f(-3) \). Using \( f(x)=x - 1 \), \( f(-3)=-3-1=-4

eq3 \).

• Option D: \( g(x)=f(x)+4 \). The y - intercept of \( g(x) \) is \( f(0)+4=-1 + 4 = 3 \), which matches the required y - intercept.

Answer:

D. \( g(x)=f(x)+4 \)