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

1 Answer
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=...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.