To determine if each linear function is increasing or decreasing, we need to look at the slope of each function. The slope is represented by the coefficient of [tex]\(x\)[/tex] in the linear equation of the form [tex]\(f(x) = mx + c\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept.
1. For the function [tex]\( f(x) = -8x + 1 \)[/tex]:
- The slope ([tex]\(m\)[/tex]) is [tex]\(-8\)[/tex].
- Since the slope is negative, this function is decreasing.
2. For the function [tex]\( f(x) = 9x - 4 \)[/tex]:
- The slope ([tex]\(m\)[/tex]) is [tex]\(9\)[/tex].
- Since the slope is positive, this function is increasing.
3. For the function [tex]\( f(x) = -x - 2 \)[/tex]:
- The slope ([tex]\(m\)[/tex]) is [tex]\(-1\)[/tex].
- Since the slope is negative, this function is decreasing.
4. For the function [tex]\( f(x) = x - 6 \)[/tex]:
- The slope ([tex]\(m\)[/tex]) is [tex]\(1\)[/tex].
- Since the slope is positive, this function is increasing.
5. For the function [tex]\( f(x) = -4x - 9 \)[/tex]:
- The slope ([tex]\(m\)[/tex]) is [tex]\(-4\)[/tex].
- Since the slope is negative, this function is decreasing.
6. For the function [tex]\( f(x) = -10x + 2 \)[/tex]:
- The slope ([tex]\(m\)[/tex]) is [tex]\(-10\)[/tex].
- Since the slope is negative, this function is decreasing.
Therefore, the increasing and decreasing classification for each function is as follows:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Function} & \text{Classification} \\
\hline
f(x) = -8x + 1 & \text{decreasing} \\
f(x) = 9x - 4 & \text{increasing} \\
f(x) = -x - 2 & \text{decreasing} \\
f(x) = x - 6 & \text{increasing} \\
f(x) = -4x - 9 & \text{decreasing} \\
f(x) = -10x + 2 & \text{decreasing} \\
\hline
\end{array}
\][/tex]
