Question

select the correct answer. consider the graph of function f below. graph of a line on a coordinate plane the function g is a transformation of f. if g has a y - intercept at - 1, which of the following functions could represent g? a. $g(x)=f(x)-1$ b. $g(x)=f(x)+4$ c. $g(x)=f(x + 4)$ d. $g(x)=f(x - 1)$

Explanation:

Step1: Find y-intercept of f(x)

From the graph, when \( x = 0 \), the y - value (y - intercept) of \( f(x) \) is \( - 5 \) (by looking at the point where the line crosses the y - axis).

Step2: Analyze each option

• Option A: \( g(x)=f(x)-1 \). The y - intercept of \( g(x) \) is the y - intercept of \( f(x) \) minus 1. So, \( - 5-1=-6

eq - 1 \).

• Option B: \( g(x)=f(x) + 4 \). The y - intercept of \( g(x) \) is the y - intercept of \( f(x) \) plus 4. So, \( - 5 + 4=-1 \), which matches the given y - intercept of \( g(x) \).
• Option C: \( g(x)=f(x + 4) \). This is a horizontal shift. The y - intercept of \( f(x+4) \) occurs when \( x = 0 \), so we find \( f(4) \). From the slope of \( f(x) \) (slope \( m=\frac{\Delta y}{\Delta x}=\frac{-5 - 7}{0+7}=- \frac{12}{7}\)? Wait, better to use two points. Let's take two points on \( f(x) \): when \( x=-3 \), \( y = 1 \); when \( x = 0 \), \( y=-5 \). Slope \( m=\frac{-5 - 1}{0+3}=-2 \). Equation of \( f(x) \): \( y-1=-2(x + 3)\), \( y=-2x-6 + 1=-2x-5 \). So \( f(x)=-2x-5 \). Then \( f(x + 4)=-2(x + 4)-5=-2x-8 - 5=-2x-13 \). Y - intercept is \( - 13

eq - 1 \).

• Option D: \( g(x)=f(x - 1) \). \( f(x-1)=-2(x - 1)-5=-2x+2 - 5=-2x-3 \). Y - intercept is \( - 3

eq - 1 \).

Answer:

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