Question

How do you graph `16(x-9)=(y+9)^2`?

Answer 1

It would be a sideways parabola.
graph{x=((y+9)^2)/16+9 [-56.3, 91.86, -45.3, 28.74]}

Explanation:

There are two ways to interpret this relation between `x` and `y`.

1. Let y be the independent variable and x be the dependent variable. Write the equation as `y=...x`
2. Rewrite the equation so `x` is the dependent variable: `x=...y` then rotate the graph so that `y` is the dependent variable again.

Option 1 is easier to understand but harder to rewrite this equation for. Option 2 takes a new approach at looking at the nature of graphs. We'll be going for Option 2.

We know that in the Cartesian plane, `x` is the horizontal, independent, axis and `y` is the vertical, dependent, axis. This shows that "If I have `x`, I can find `y`".

What if we rewrite this such that "If I have `y`, I can find `x`"?

So, how does this relate to your question?
In `16(x-9)=(y+9)^2`, we can rewrite the equation to be the function of `y`.

At this point, I believe that using function notation is easier to declare which variable is independent and which is dependent. If you are unfamiliar, this is a quick overview:
In `f(x)`, `f(x)` is the dependent variable and `x` is the independent variable.
In `h(y)`, `h(y)` is the dependent and the `y` would be the independent.

Consider your equation rewritten using function notation:
`16(x-9)=(f(x)+9)^2`

And then we switch `x` and `f(x)`:
`16(f(x)-9)=(x+9)^2`
`f(x)=(x+9)^2/16+9`
We'll call this rewritten function `g(x)` from now on.

When we graph this out with the axis switched, we get a parabola as expected:

graph{y=(x+9)^2/16+9 [-122.1, 115.1, -2.4, 116.3]}

However, we must switch the roles of `x` and `f(x)` back, which means switching the vertical and horizontal axis. This is done by rotating everything clockwise 90 degrees about the origin.

graph{x=((y+9)^2)/16+9 [-56.3, 91.86, -45.3, 28.74]}

I'd like to leave off at this property:

If `f(x)=x^2`
And `x=g(x)^2`

Then `g(x)` is `f(x)` rotated 90 degrees clockwise about the origin.