Let's analyze the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
First, we are given the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = -6x + 3 \][/tex]
From this equation, we can identify the slope and the y-intercept.
- Slope of [tex]\( g(x) \)[/tex]: The slope is the coefficient of [tex]\( x \)[/tex], which is [tex]\(-6\)[/tex].
- y-intercept of [tex]\( g(x) \)[/tex]: The y-intercept is the constant term, which is [tex]\(3\)[/tex].
Next, we need to determine the slope and y-intercept of the function [tex]\( f(x) \)[/tex]. Based on our previous analysis, we have:
- Slope of [tex]\( f(x) \)[/tex]: [tex]\(-6\)[/tex]
- y-intercept of [tex]\( f(x) \)[/tex]: [tex]\(-2\)[/tex]
Now let's compare these values.
1. Slopes:
- Slope of [tex]\( f(x) \)[/tex]: [tex]\(-6\)[/tex]
- Slope of [tex]\( g(x) \)[/tex]: [tex]\(-6\)[/tex]
- Comparison: The slopes are the same.
2. y-intercepts:
- y-intercept of [tex]\( f(x) \)[/tex]: [tex]\(-2\)[/tex]
- y-intercept of [tex]\( g(x) \)[/tex]: [tex]\(3\)[/tex]
- Comparison: The y-intercepts are different.
Given these comparisons:
- The slopes are the same, but the y-intercepts are different.
Therefore, the correct option is:
[tex]\[ \boxed{\text{C. The slopes are the same but the y-intercepts are different.}} \][/tex]
