Question

two linear functions are graphed in the coordinate plane.
if $g(x) = f(x) + k$, then what is the value of $k$ ?

Explanation:

Step1: Find y-intercepts of \( g(x) \) and \( f(x) \)

The y - intercept of a linear function \( y = mx + b \) is the value of \( y \) when \( x = 0 \). For \( g(x) \), looking at the graph, when \( x = 0 \), \( g(0)=4 \) (since the line \( g(x) \) crosses the y - axis at \( (0,4) \)). For \( f(x) \), when \( x = 0 \), \( f(0)= - 4 \) (since the line \( f(x) \) crosses the y - axis at \( (0, - 4) \)).

Step2: Use \( g(x)=f(x)+k \) at \( x = 0 \)

Substitute \( x = 0 \) into the equation \( g(x)=f(x)+k \). We get \( g(0)=f(0)+k \). We know \( g(0) = 4 \) and \( f(0)=-4 \). So, \( 4=-4 + k \).

Step3: Solve for \( k \)

To solve for \( k \), add 4 to both sides of the equation \( 4=-4 + k \). \( k=4 + 4=8 \)? Wait, no, wait. Wait, maybe I made a mistake in y - intercepts. Wait, let's re - check. Wait, looking at the graph again. Wait, the upper line is \( g(x) \), lower is \( f(x) \). Let's find the y - intercepts correctly. Let's see, for \( g(x) \), when \( x = 0 \), the y - coordinate is 4? Wait, no, maybe the grid. Wait, each grid is 1 unit. Let's see, the line \( g(x) \): when \( x = 0 \), it's at \( y = 4 \)? Wait, no, maybe the y - intercept of \( g(x) \) is 4 and \( f(x) \) is - 4? Wait, no, let's take another approach. The vertical shift between \( g(x) \) and \( f(x) \). Since \( g(x)=f(x)+k \), \( k \) is the vertical shift. The distance between the y - intercepts of \( g(x) \) and \( f(x) \) is \( k \). Let's find the y - intercepts. For \( g(x) \), let's pick a point. Let's say when \( x = 0 \), \( g(0)=4 \) (from the graph, the upper line crosses y - axis at \( (0,4) \)). For \( f(x) \), when \( x = 0 \), \( f(0)=-4 \) (lower line crosses y - axis at \( (0, - 4) \)). Then \( g(x)=f(x)+k \), so at \( x = 0 \), \( 4=-4 + k \), so \( k = 8 \)? Wait, no, that can't be. Wait, maybe I mixed up the lines. Wait, let's check the slopes. Both lines are parallel, so same slope. The equation \( g(x)=f(x)+k \) means that \( g(x) \) is a vertical shift of \( f(x) \). So the difference in their y - values at the same x is \( k \). Let's take \( x = 0 \). \( g(0)-f(0)=k \). If \( g(0) = 4 \) and \( f(0)=-4 \), then \( k=4-(-4)=8 \)? Wait, no, wait, maybe the y - intercept of \( g(x) \) is 4 and \( f(x) \) is - 4? Wait, no, maybe I had the y - intercepts wrong. Wait, let's look at the graph again. Wait, the upper line (g(x)): when x = 0, it's at y = 4? Wait, no, maybe the y - intercept of g(x) is 4 and f(x) is - 4? Wait, no, let's count the units. From f(x)'s y - intercept (0, - 4) to g(x)'s y - intercept (0,4), the distance is 8 units up. So \( k = 8 \)? Wait, no, wait, maybe the y - intercept of g(x) is 4 and f(x) is - 4, so \( g(x)=f(x)+8 \), so \( k = 8 \)? Wait, but let's check with another point. Let's take x = 8. For g(x), when x = 8, y = 0 (since it crosses the x - axis at (8,0)). For f(x), when x = 8, y=-8 (since the slope is - 1, because from (0, - 4) to (8, - 12)? Wait, no, maybe slope is - 1. Wait, slope of a line is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For g(x), from (0,4) to (8,0), slope \( m=\frac{0 - 4}{8 - 0}=\frac{-4}{8}=-\frac{1}{2} \). For f(x), from (0, - 4) to (8, - 8), slope \( m=\frac{-8+4}{8 - 0}=\frac{-4}{8}=-\frac{1}{2} \). Now, \( g(x)=f(x)+k \). Let's substitute x = 8. \( g(8)=0 \), \( f(8)=-8 \). So \( 0=-8 + k \), so \( k = 8 \). Yes, that's correct. So \( k = 8 \)? Wait, no, wait, when x = 0, g(0)=4, f(0)=-4. Then \( 4=-4 + k \), so \( k = 8 \). So the value of k is 8.

Answer:

8