Question

Consider the graph of f(x) given below.

A curve labeled f of x declines through the points (negative 4, 7), (negative 3, 6), (negative 2, 5), (0, 3), (2, 1), (3, 0), (4, negative 1), (5, negative 2), (6, negative 3) and (7, negative 4) on the x y coordinate plane.

The function g(x) is a transformation of f(x). If g(x) has a y-intercept of -2, which of the following functions could represent g(x)?

A.
g(x) = f(x - 5)
B.
g(x) = f(x) - 5
C.
g(x) = f(x + 2)
D.
g(x) = f(x) - 2

Answer 1

To find which function \( g(x) \) has a y-intercept of -2, we need to evaluate the y-intercept of each option based on the transformation to \( f(x) \).

Recall that the y-intercept of a function \( g(x) \) is \( g(0) \).

1. **Option A: \( g(x) = f(x - 5) \)**
To find the y-intercept:
\[
g(0) = f(0 - 5) = f(-5)
\]
Since \( f(-5) \) is not given in the points listed, we cannot determine if \( g(0) = -2 \).

2. **Option B: \( g(x) = f(x) - 5 \)**
To find the y-intercept:
\[
g(0) = f(0) - 5 = 3 - 5 = -2
\]
This option satisfies the condition \( g(0) = -2 \).

3. **Option C: \( g(x) = f(x + 2) \)**
To find the y-intercept:
\[
g(0) = f(0 + 2) = f(2) = 1
\]
This does not satisfy the condition since \( g(0) \neq -2 \).

4. **Option D: \( g(x) = f(x) - 2 \)**
To find the y-intercept:
\[
g(0) = f(0) - 2 = 3 - 2 = 1
\]
This does not satisfy the condition since \( g(0) \neq -2 \).

Based on this analysis, the only function \( g(x) \) that has a y-intercept of -2 is:

**B. \( g(x) = f(x) - 5 \)**