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Consider the two functions $f(x)=ax^2$ and $g(x)=bx^2$. Using this transformation form $T(x,y)=(cx,cy)$, find a scale change that maps $f(x)$ onto $g(x)$

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Temporarily, let $s=cx$ and $t=cy$. Then $x=\dfrac{s}{c}$ and $y=\dfrac{t}{c}$. Inserting in the equation $y=ax^2$ we get $$\frac{t}{c}=a\frac{s^2}{c^2}.$$ We want this to read $t=bs^2$. For that, we need $\dfrac{a}{c}=b$, or equivalently $c=\dfrac{a}{b}$.

So now our equation reads $t=bs^2$. Replace the letter $s$ by $x$, and $t$ by $y$.

Remark: The result has an interesting geometric interpretation: all parabolas look alike, just like all circles look alike.

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