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I don't understand how to get this answer.

$1)$ Suppose that the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f$. $$y=-2f(x)$$

Answer: Reflect in the $x$-axis and stretch vertically by a factor of $2$.

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  • $\begingroup$ What does this have to do with translations? Don't you mean reflections? $\endgroup$
    – Tyler
    Nov 3, 2013 at 4:49

2 Answers 2

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The reflection over the $x$-axis is due to the $-$ out front, which multiplies all the $y$-values by $-1$. The vertical stretch factor of $2$ multiplies all the $y$-values by $2,$ which stretches the graph out vertically. It doesn't matter whether we stretch or reflect first. The end result--multiplying all the $y$-values of the graph of $f(x)$ by $-2$--is the same.

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I think a picture makes this a bit clearer.

enter image description here

Here are the graphs of four functions (they're half-ellipses). The blue one we'll call $\color{blue}{f(x)}$. If we change the sign (which is the same as multiplying it by $-1$), we will flip the graph over the $x$-axis, which gives us the green function, $\color{limegreen}{-f(x)}$. Now, what if we multiply $\color{blue}{f(x)}$ by $2$? Well, that will stretch out the graph because every value we got from the original function will be multiplied by $2$, which will put it twice as high on the graph. So, multiplying by $2$ gives us the purple function, $\color{purple}{2f(x)}$. Now, let's do the same thing we did the get the green function. If we multiply $\color{purple}{2f(x)}$ by $-1$, we flip the graph and get the yellow function, $\color{goldenrod}{-2f(x)}$.

So, you can see how $\color{goldenrod}{-2f(x)}$ can be obtained by reflecting $\color{blue}{f(x)}$ in the $x$-axis and stretching it vertically by a factor of $2$.

By the way, a translation (at least the way I learned it) is a "slide," like this:

enter image description here

A better, more general word for what we did above would probably be transformation.

I hope this helps :)

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