To reflect the function [tex]\(f(x) = \frac{1}{2}x + 8\)[/tex] in the x-axis, we need to understand the effect of this transformation. Reflecting a function in the x-axis involves multiplying the entire function by [tex]\(-1\)[/tex]. This means that every output value of the function is inverted.
Let's go through the steps of this transformation:
1. Original Function: The original function is given by:
[tex]\[
f(x) = \frac{1}{2}x + 8
\][/tex]
2. Reflection in the x-axis: To reflect this function in the x-axis, we multiply the entire function by [tex]\(-1\)[/tex]:
[tex]\[
g(x) = -f(x)
\][/tex]
3. Apply the Multiplication: Substitute the original function [tex]\(f(x)\)[/tex] into this equation:
[tex]\[
g(x) = -\left(\frac{1}{2}x + 8\right)
\][/tex]
4. Distribute the Negative Sign: Distribute the [tex]\(-1\)[/tex] across the terms inside the parentheses:
[tex]\[
g(x) = -\frac{1}{2}x - 8
\][/tex]
Thus, the transformed function is:
[tex]\[
g(x) = -\frac{1}{2}x - 8
\][/tex]
To summarize, reflecting the function [tex]\(f(x) = \frac{1}{2}x + 8\)[/tex] in the x-axis results in the new function [tex]\(g(x) = -\frac{1}{2}x - 8\)[/tex].
If we identify the slope and y-intercept of the transformed function:
- The slope of the new function [tex]\(g(x)\)[/tex] is [tex]\(-0.5\)[/tex]
- The y-intercept of the new function [tex]\(g(x)\)[/tex] is [tex]\(-8\)[/tex]
Therefore, the detailed step-by-step solution yields that the transformed function through reflection in the x-axis is [tex]\(g(x) = -\frac{1}{2}x - 8\)[/tex].
