How to make a parabola tilt.

Question

Given a parabola $Y=X^2,$ how can I modify the equation so the parabola tilts away from it’s vertical orientation? For example, a solar collector might have a tilt of $23.5$ degrees. How would this look as a polynomial?

Answer 1

The parabola $y=x^2$ has a vertical axis. I will construe your $23.5^\circ$ to mean $23.5^\circ$ from the vertical.

Draw the $x$-axis pointing to the right and the $y$-axis pointing upward.

Then draw a $u$-axis pointing in a direction $23.5^\circ$ clockwise from the $x$-axis and a $v$-axis $23.5^\circ$ clockwise from the $y$-axis.

You need the equation of your rotated parabola to be $v=u^2.$

We will need this: $$ \tag 1 \begin{align} u & = (\cos23.5^\circ)x-(\sin23.5^\circ) y \\ v & = (\sin23.5^\circ)x +(\cos23.5^\circ) y \end{align} $$

So $v=u^2$ becomes $$ (\sin23.5^\circ)x +(\cos23.5^\circ) y = \Big( (\cos23.5^\circ)x-(\sin23.5^\circ) y \Big)^2. $$

For now I've left the derivation of $(1)$ as an exercise, but if necessary you can ask about that too.

Answer 2

If your parabola is given by:

$$ -x\sin a+y\cos a\ =\left(\ x\cos a+y\sin a-h\right)^2+k $$

changing $h$ will move it along $x$ axis, changing $k$ will move it along the $y$ axis. Altering $a$ will tilt it.

See demo: https://www.desmos.com/calculator/hguanwbkbu