This lesson is optional in the Precalculus course.
This lesson assumes knowledge of
equations of ellipses in standard form: foci on the $\,x$axis.
You may find yourself thinking that an ellipse looks like a
‘deformed circle’:
you know, just grab the ‘ends’ and stretch or compress the circle.
And—you'd be correct, as this lesson shows!
By using earlier work on horizontal and vertical stretching/shrinking of graphs, we get the equation $\displaystyle\,\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\,$ easily, as follows:

$x^2 + y^2 = 1$ horizontal stretch, in red: $\displaystyle\frac{x^2}{a^2} + y^2 = 1\,$, for $\,a > 1$ horizontal shrink, in green: $\displaystyle\frac{x^2}{a^2} + y^2 = 1\,$, for $\,0 < a < 1$ circle of radius $\,1\,$, in blue: $x^2 + y^2 = 1$ vertical stretch, in red: $\displaystyle x^2 + \frac{y^2}{b^2} = 1\,$, for $\,b > 1$ vertical shrink, in green: $\displaystyle x^2 + \frac{y^2}{b^2} = 1\,$, for $\,0 < b < 1$ 
People who have thoroughly studied
horizontal and vertical stretching/shrinking
may be a bit perplexed
by the treatment of the transformations involving $\,y\,$ in this section.
If this is you, then keep reading!
In that earlier section, we were dealing
with a very specific type of equation involving $\,x\,$ and $\,y\,$: an equation of the form $\,y = f(x)\,$.
Such an equation has $\,y\,$ all by itself on one side; the other side involves only $\,x\,$.
For equations of the form $\,y = f(x)\,$, it was easier to treat transformations involving $\,x\,$ and $\,y\,$ differently.
We'll quickly review those differences next.
Transformations involving $y$ for equations of the form $\,y = f(x)\,$:
Sometimes you need to apply graphical transformations to equations
that can't be put in the form $\,y = f(x)\,$.
If the graph of an equation doesn't pass a vertical line test, then it can't be put in the form $\,y = f(x)\,$.
The ellipse equation $\displaystyle\,\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\,$
is such an equation.
Its graph doesn't pass the vertical line test.
Here, it is possible to solve for $\,y\,$, but it introduces an annoying square root and ‘plus or minus’ sign.
For example:
If we want every $y$value in a graph to be multiplied by $\,3\,$,
then replace every $\,y\,$ in the original equation by $\,y\,$ divided by $\,3\,$.
This more general technique certainly works in the ‘$\,y = f(x)\,$’ case, but isn't needed:
EXAMPLE:
Start with the graph of $\,y = x^2\,$.
Multiply each $y$value of points on the graph by $\,3\,$ (a vertical stretch).
What is the equation of the resulting graph?
ANSWER:
$\displaystyle\,\frac{y}{3} = x^2\,$
Replace every $\,y\,$ by $\,\frac{y}{3}\,$.
Of course, this gives the same result—just multiply
both sides by $\,3\,$ to get $\,y = 3x^2\,$!
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
IN PROGRESS 