X Axis Dilation 91333C

1. Let's start by understanding the role of $a$ in transformations of functions. 2. When we write a function transformation like $y = f(ax)$, the parameter $a$ affects the $x$-axis by stretching or compressing the graph horizontally. 3. Specifically, if $|a| > 1$, the graph compresses horizontally by a factor of $\frac{1}{|a|}$. 4. If $0 < |a| < 1$, the graph stretches horizontally by a factor of $\frac{1}{|a|}$. 5. Choosing $a=1$ means no horizontal dilation; the graph remains unchanged in the $x$ direction. 6. If there was a dilation in the $x$ axis, i.e., $a \neq 1$, the function's input values are scaled, changing the shape's width. 7. For example, if $a=2$, the graph compresses horizontally by $\frac{1}{2}$, making it twice as narrow. 8. Conversely, if $a=\frac{1}{2}$, the graph stretches horizontally by a factor of 2, making it twice as wide. 9. This dilation affects how quickly the function changes with respect to $x$. 10. In summary, choosing $a=1$ simplifies analysis by avoiding horizontal dilation, but if $a \neq 1$, the graph is horizontally scaled accordingly.