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


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

You are watching: How to tilt a parabola

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

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

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

We will certainly require this:$$ ag 1eginalignu & = (cos23.5^circ)x-(sin23.5^circ) y \v & = (sin23.5^circ)x +(cos23.5^circ) yendalign$$

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 crucial you have the right to ask around that as well.


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answered May 5 "18 at 19:44
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Michael HardyMichael Hardy
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If your parabola is given by:

$$ -xsin a+ycos a =left( xcos a+ysin a-h ight)^2+k $$

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

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


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answered May 5 "18 at 19:08
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ITAITA
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