## Function Notation

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 \(f(x) + 2\).

## Functions in Geometry and 3D

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.

## Learn More

- Explore Functions Example Graph
- Generalizing "for" Lists and Intervals
- Lists
- Tables
- Vectors and Point Operations

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