Functions

You can use function notation to make connections between expressions, tables, and other mathematical objects. Evaluate a function at a specific value, attach a moveable point to a curve, create input and output tables, and more.

Watch the video for an introduction to function notation, and read on to learn more.

Use function notation to evaluate functions for a certain value. For example, if \(f(x) = x^2+3x\), you can then evaluate \(f(3) = 18\).

You can also define a function with more than one variable. For example, if \(g(x,y) = x^2+3y\), \(g(10,3)\) will evaluate to \(10^2+3(3)=109\).

Once you've defined a function, you can use it in expressions or within other functions. For example, if you have \(f(x) = x^2 + 3x\), you can add \(2\) to the function to perform a vertical shift using the expression \(f(x) + 2\).

You can also try using function notation with a slider to trace a function.

For example, if \(f(x) = \sin(x)\) you have a point \((a,f(a))\) in a new line, the calculator 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,\(180)\).

Because transformations are defined as functions, you can use function notation to reference them elsewhere. Try applying \(T_{1}\) to a different object or composing multiple transformations like in the example. To apply a function to a geometric object while focused in the function, hold Shift and click on the object.

You can also apply a function to points. Consider the example \(f(a) = \frac{1}{2}a + (0,0,2)\). 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.