Functions

Function notation can be used to make meaningful connections between expressions, tables, and other mathematical objects. Use function notation to evaluate a function at a specific value, attach a moveable point to a curve, create input and output tables, and more. Get started with the video on the right, then dive deeper with the resources and challenges below.

Evaluate functions for a certain value using function notation. For example, if you set \(f(x) = x^2+3x\), you can then evaluate \(f(3) = 18\). You can also define a function with more than one variable. Try typing \(g(x,y) = x^2+3y\) and note that \(g(10,3)\) evaluates to \(10^2+3(3)=109\). Click on any of the images to open the graph and explore.

Once a function has been defined, it can be used in expressions or within other functions. With \(f(x) = x^2 + 3x\), try adding \(2\) to the function (performing a vertical shift) by typing \(f(x) + 2\).

Try using function notation with a slider to trace a function. For example, set \(f(x) = \sin(x)\), and then type \((a,f(a))\) in a new line. Desmos will create an adjustable slider for the variable \(a\). As you drag the slider, the point will trace along the curve.

In the Desmos Geometry Tool, transformations are defined as functions. As an example, trying using the toolbar to define a \(180°\) rotation. When you pull the transformation down from the token navigator, you will see it defined as a function \(T_1(x) =\) rotate\((x\),point,\(180)\).

Since transformations are defined as functions, you can use function notation to use them repeatedly. Try applying T1 to a different object or composing multiple transformations like in the example.

Open the example graph by clicking on the GIF to see these transformations on a triangle and try out your own transformations.

You can also apply a function to points! Let’s look at the function \(f(a) = \frac{1}{2}a + (0,0,2)\). In this example, each point \((a)\) is scaled by a factor of \(\frac{1}{2}\) before it’s translated up \(2\) units on the \(z\) axis.

Explore this function by clicking on the image.