Let's compare the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] by analyzing their slopes and y-intercepts.
1. Function [tex]\( f(x) = \frac{1}{4}x - 1 \)[/tex]:
- The slope of the function [tex]\( f(x) \)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
- The y-intercept of [tex]\( f(x) \)[/tex] is [tex]\(-1\)[/tex].
2. Function [tex]\( g(x) = \frac{1}{2}x - 2 \)[/tex]:
- The slope of the function [tex]\( g(x) \)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
- The y-intercept of [tex]\( g(x) \)[/tex] is [tex]\(-2\)[/tex].
### Step-by-Step Comparison:
- Slopes:
- For [tex]\( f(x) \)[/tex], the slope is [tex]\(\frac{1}{4}\)[/tex].
- For [tex]\( g(x) \)[/tex], the slope is [tex]\(\frac{1}{2}\)[/tex].
When comparing [tex]\(\frac{1}{4}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]:
- [tex]\(\frac{1}{2}\)[/tex] is greater than [tex]\(\frac{1}{4}\)[/tex].
- A greater slope means that the line is steeper. Therefore, the line of [tex]\( g(x) \)[/tex] is steeper than the line of [tex]\( f(x) \)[/tex].
- Y-intercepts:
- For [tex]\( f(x) \)[/tex], the y-intercept is [tex]\(-1\)[/tex].
- For [tex]\( g(x) \)[/tex], the y-intercept is [tex]\(-2\)[/tex].
When comparing [tex]\(-1\)[/tex] and [tex]\(-2\)[/tex]:
- [tex]\(-2\)[/tex] is less than [tex]\(-1\)[/tex].
- A lower y-intercept means that the line crosses the y-axis at a lower point. Therefore, the y-intercept of [tex]\( g(x) \)[/tex] is lower than the y-intercept of [tex]\( f(x) \)[/tex].
### Conclusion:
The line of [tex]\( g(x) \)[/tex] is steeper and has a lower y-intercept than the line of [tex]\( f(x) \)[/tex].
The correct answer is:
The line of [tex]\( g(x) \)[/tex] is steeper and has a lower y-intercept.
