## Warm-up solution

The graph of the function $y=\dfrac{1}{x^2}$ is stretched as shown in this diagram.

If the graph is stretched by a factor of $b$ in the $x$-direction (as shown), what is the equation of the resulting graph?

The original graph has equation $y=f(x)$, where $f(x)=\dfrac{1}{x^2}$.
When we stretch the graph by a factor of $b$ in the $x$-direction, a point with coordinates $(x,f(x))$ is stretched to the point with coordinates $(bx,f(x))$, as shown in the following diagram. If we write $X=bx$ for the new $x$-coordinate, then the new point has coordinates $(X,Y)=(X,f(X/b))$, so the new graph has equation $Y=f(X/b)$, or $y=f(x/b)$ using lower-case letters.
So the equation of the stretched graph is $y=f(x/b)=\dfrac{1}{(x/b)^2}$. Multiplying the numerator and denominator by $b^2$ gives the simpler form $y=\dfrac{b^2}{x^2}$.
And if, instead, the original graph is stretched by a factor of $c$ in the $y$-direction, what is the equation of the resulting graph?
This time, the point $(x,f(x))$ is stretched to the point with coordinates $(x,cf(x))$, as shown in the following diagram. Writing the coordinates of the new point again as $(X,Y)=(x,cf(x))$, we find the new graph has equation $Y=cf(X)$, as $X=x$, or $y=cf(x)$ using lower-case letters.
Therefore the equation of the stretched graph is $y=\dfrac{c}{x^2}$.