To match each transformation of function [tex]\( f \)[/tex] to a notable feature of the transformed function, we'll consider how each transformation affects the function and match it with the appropriate feature. Here are the detailed steps:
1. Understanding [tex]\( h(x) = f(x) + 2 \)[/tex]:
- This transformation shifts the function [tex]\( f \)[/tex] vertically up by 2 units.
- If the original [tex]\( f(x) \)[/tex] had a y-intercept at (0,2), then [tex]\( h(x) = f(x) + 2 \)[/tex] will have a y-intercept 2 units higher.
- Therefore, the y-intercept will be at (0,4).
Matching:
- [tex]\( h(x) = f(x) + 2 \)[/tex] matches with [tex]\( y \)[/tex]-intercept at (0,4).
2. Understanding [tex]\( j(x) = f(x + 2) \)[/tex]:
- This transformation shifts the function [tex]\( f \)[/tex] horizontally to the left by 2 units.
- The behavior of the function as [tex]\( x \)[/tex] changes could be shifted similarly. If the function [tex]\( f \)[/tex] decreases as [tex]\( x \)[/tex] increases, the shift to the left won't change the decreasing nature of the function.
Matching:
- [tex]\( j(x) = f(x + 2 \)[/tex] matches with [tex]\( \)[/tex]function decreases as [tex]\( x \)[/tex] increases.
3. Understanding [tex]\( m(x) = -f(x) \)[/tex]:
- This transformation reflects the function [tex]\( f \)[/tex] over the x-axis.
- If [tex]\( f(x) \)[/tex] has an asymptote, reflecting it over the x-axis won't affect the x-position of the asymptote, but its y-value may change based on the nature of [tex]\( f \)[/tex].
Matching:
- [tex]\( m(x) = -f(x) \)[/tex] matches with [tex]\( \)[/tex]asymptote of [tex]\( y = 2 \)[/tex].
4. Understanding [tex]\( g(x) = 2 f(x) \)[/tex]:
- This transformation stretches the function [tex]\( f \)[/tex] vertically by a factor of 2.
- If the original [tex]\( f(x) \)[/tex] had a y-intercept at (0,1), then [tex]\( g(x) = 2 f(x) \)[/tex] will double the y-intercept value to (0,2).
Matching:
- [tex]\( g(x) = 2 f(x) \)[/tex] matches with [tex]\( y \)[/tex]-intercept at (0,2).
Putting it all together, the correct matches are:
[tex]\[
\begin{align*}
\boxed{h(x) = f(x) + 2} \ &\longrightarrow \ y \text{-intercept at } (0,4), \\
\boxed{j(x) = f(x + 2)} \ &\longrightarrow \ \text{function decreases as } x \text{ increases}, \\
\boxed{m(x) = -f(x)} \ &\longrightarrow \ \text{asymptote of } y = 2, \\
\boxed{g(x) = 2 f(x)} \ &\longrightarrow \ y \text{-intercept at } (0,2).
\end{align*}
\][/tex]
