Making New Functions

Algebra students usually learn at some point in their studies that you can dilate a function like \(\mathtt{f(x)=x^2}\) by multiplying it by a constant value. Usually the multiplier is written as \(\mathtt{A}\), so you get \(\mathtt{f(x)=Ax}\), which can be \(\mathtt{f(x)=2x^2}\) or \(\mathtt{f(x)=3x^2}\) or \(\mathtt{f(x)=\frac{1}{2}x^2}\), and so on.

What we focus on at the beginning is how this change affects the graph of the function—and, importantly, how we can consistently describe that change as it applies to any function.

So, for example, the brightest (orange-brown) curve in the image at the right represents \(\mathtt{f(x)=x^2}\), or \(\mathtt{f(x)=1x^2}\). And when we increase the A-value, from 2 to 3, the curve gets narrower. Decreasing the A-value, to \(\mathtt{\frac{1}{2}}\) for example, causes the curve to get wider. The same kind of “squishing” happens with every function type. (Check out this article by Better Explained for a really nice explanation.)

Making New Functions

We can also make higher-degree functions from lower-degree functions using dilations. The only change we make to the process of dilation shown above is that now we multiply each point of a function by a non-constant value. More specifically, we can multiply the y-coordinate of each point by its x-coordinate to get its new y-coordinate.

At the left, we transform the constant function \(\mathtt{f(x)=5}\) into a higher-order function in this way. By multiplying the y-coordinate of each original point by its x-coordinate, we change the function from \(\mathtt{f(x)=5}\) to \(\mathtt{f(x)=(x)(5)}\), or \(\mathtt{f(x)=5x}\). Another multiplication of all the points by x would get us \(\mathtt{f(x)=5x^2}\). In that case, you can see that all the points to the left of the y-axis have to reflect across the x-axis, since each y-coordinate would be a negative times a negative.

Another idea that becomes clearer when working with non-constant dilations in this way is that zeros start to make a little more sense.

Try it with other dilations (say, \(\mathtt{x \cdot (x+3)}\) or even \(\mathtt{x \cdot (x-1)^2)}\)) and pay attention to what happens to those points that wind up getting multiplied by 0.

Published by

Josh Fisher

Instructional designer, software development in K-12 mathematics education.