Describe how to transform the graph of $f$ into the graph of g. Sketch the graphs by hand and support your answer with a grapher.
\[
\begin{array}{l}
f(x)=e^{x} \\
g(x)=e^{-10 x}
\end{array}
\]
To transform the graph of $\mathrm{f}$ into the graph of $\mathrm{g}$,
Answer
So, to transform the graph of \(f(x) = e^x\) into the graph of \(g(x) = e^{-10x}\), you would first reflect the graph of \(f(x)\) in the y-axis, and then compress it horizontally by a factor of 10.
Step 1 ：The transformation from the graph of \(f(x) = e^x\) to \(g(x) = e^{-10x}\) involves two steps.
Step 2 ：The first step is a reflection. The negative sign in the exponent of \(g(x) = e^{-10x}\) indicates a reflection of the graph of \(f(x) = e^x\) in the y-axis. This means that the graph of \(g(x)\) will be a mirror image of the graph of \(f(x)\) with respect to the y-axis.
Step 3 ：The second step is a horizontal compression. The factor of 10 in the exponent of \(g(x) = e^{-10x}\) indicates a horizontal compression of the graph of \(f(x) = e^x\) by a factor of 10. This means that the graph of \(g(x)\) will be 'squeezed' horizontally towards the y-axis by a factor of 10 compared to the graph of \(f(x)\).
Step 4 ：So, to transform the graph of \(f(x) = e^x\) into the graph of \(g(x) = e^{-10x}\), you would first reflect the graph of \(f(x)\) in the y-axis, and then compress it horizontally by a factor of 10.
