1. Let's start by stating the problem: You want to understand how to identify horizontal and vertical dilations and translations in a function. 2. The general form of transformations for a function $f(x)$ is: - Horizontal dilation and translation: $f(b(x - h))$ - Vertical dilation and translation: $a f(x) + k$ 3. Important rules: - Horizontal transformations affect the input $x$ inside the function. - Vertical transformations affect the output of the function. 4. Horizontal dilation (stretch/compression) is controlled by $b$: - If $|b| > 1$, the graph compresses horizontally. - If $0 < |b| < 1$, the graph stretches horizontally. 5. Horizontal translation (shift) is controlled by $h$: - The graph shifts right by $h$ units if $h > 0$. - The graph shifts left by $|h|$ units if $h < 0$. 6. Vertical dilation (stretch/compression) is controlled by $a$: - If $|a| > 1$, the graph stretches vertically. - If $0 < |a| < 1$, the graph compresses vertically. 7. Vertical translation (shift) is controlled by $k$: - The graph shifts up by $k$ units if $k > 0$. - The graph shifts down by $|k|$ units if $k < 0$. 8. Summary: - Look inside the function argument for horizontal changes: $b$ and $h$. - Look outside the function for vertical changes: $a$ and $k$. This way, you can identify which transformations are horizontal or vertical by their position relative to the function $f(x)$.