# Transformations of Functions Demo

Posted on 2017-05-15 by nbloomf
Tags: math

Given an equation in $$x$$ and $$y$$, if we replace all the instances of $$x$$ with $$\frac{1}{a}(x-h)$$, this has the effect of shifting the equation’s graph horizontally by $$|h|$$ units (right if $$h$$ is positive, left if negative) and stretching the graph horizontally by a factor of $$|a|$$ with a flip across the $$y$$-axis if $$a$$ is negative. Likewise, replacing all instances of $$y$$ by $$\frac{1}{b}(y-k)$$ gives a vertical shift and stretch.

The widget below is a live demonstration of these transformations. The graph of an equation is plotted on a pair of Cartesian axes, and the sliders below can be used to tweak the values of $$a$$, $$b$$, $$h$$, and $$k$$. The graph updates on the fly. You can select several different base equations using the drop-down menu. The Reset button sets $$a$$ and $$b$$ to 1 and $$h$$ and $$k$$ to 0.

Note that there is no cheating going on: the graph is drawn by checking, for each pixel, whether the equation has a solution nearby, and coloring that pixel black if so.

## Normal Equations

• Line: $x-y=0 \quad \rightarrow \quad \frac{1}{a}(x-h) - \frac{1}{b}(y-k)=0$
• Circle: $x^2 + y^2 = 1 \quad \rightarrow \quad \left(\frac{1}{a}(x-h)\right)^2 + \left(\frac{1}{b}(y-k)\right)^2 = 1$
• Quadratic: $y = x^2 \quad \rightarrow \quad \left(\frac{1}{b}(y-k)\right) = \left(\frac{1}{a}(x-h)\right)^2$
• Cubic: $y = x^3 \quad \rightarrow \quad \frac{1}{b}(y-k) = \left(\frac{1}{a}(x-h)\right)^3$

## Neat looking Equations

• Elliptic 1: $$y^2 = x^3 - 2x$$
• Elliptic 2: $$y^2 = x^3 - 2x + 2$$
• Ampersand: $$(y^2 - x^2)(x-1)(2x-3) = 4(x^2 + y^2 - 2x)^2$$
• Lemniscate: $$(x^2 + y^2)^2 = 2(x^2 - y^2)$$
• Devil’s Curve: $$y^2\left(y^2 - \frac{16}{25}\right) = x^2(x^2 - 1)$$

## What about $$t$$?

If we replace all the $$x$$s with $$x\cos(t) - y\sin(t)$$ and all the $$y$$s with $$x\sin(t) + y\cos(t)$$, this rotates the graph about the origin by $$t$$ radians (clockwise if $$t$$ is positive and counterclockwise if negative). Here the rotation happens after the shifting and stretching, which is why fiddling with the $$h$$ and $$k$$ parameters when $$t$$ is not zero don’t correspond to horizontal and vertical shifts. This demo also converts the $$t$$ parameter from degrees.