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

May 24, 2017

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 $y = \ldots x$
2. Rewrite the equation so $x$ is the dependent variable: $x = \ldots y$ 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 $16 \left(x - 9\right) = {\left(y + 9\right)}^{2}$, 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 $f \left(x\right)$, $f \left(x\right)$ is the dependent variable and $x$ is the independent variable.
In $h \left(y\right)$, $h \left(y\right)$ is the dependent and the $y$ would be the independent.

Consider your equation rewritten using function notation:
$16 \left(x - 9\right) = {\left(f \left(x\right) + 9\right)}^{2}$

And then we switch $x$ and $f \left(x\right)$:
$16 \left(f \left(x\right) - 9\right) = {\left(x + 9\right)}^{2}$
$f \left(x\right) = {\left(x + 9\right)}^{2} / 16 + 9$
We'll call this rewritten function $g \left(x\right)$ 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 $f \left(x\right)$ 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 $f \left(x\right) = {x}^{2}$
And $x = g {\left(x\right)}^{2}$

Then $g \left(x\right)$ is $f \left(x\right)$ rotated 90 degrees clockwise about the origin.