• Graph functions using a single transformation.
• Graph functions using a combination of transformations.
• Determine whether a function is even, odd, or neither from its graph.
• Describe transformations based on a function formula.
• Give the formula of a function based on its transformations.

In basic animation, an understanding of transformations (typically vector transformations) is essential for manipulating sprites. In professional animation software like Maya, After Effects, or Blender, every visual effect is created through mathematical transformations.

Consider animating a bouncing ball. The basic trajectory follows a parabolic path described by the function [latex]f(t) = -t^2+ 4[/latex], where [latex]t[/latex] represents time and [latex]f(t)[/latex] represents the ball’s height.

Next, consider a character running across the screen. The character’s basic running cycle takes 4 seconds, but you want the action to happen in just 2 seconds instead.

This is called “time remapping” – animators use horizontal compression to speed up actions without changing the motion’s shape.

Now that the character’s speed is correct, you need the movement to start 1.5 seconds later.

For the final animation transformation, a wizard character needs to grow to twice normal size while simultaneously flipping horizontally (facing the opposite direction).

Basic character model: C(x) represents the character’s outline, where x is the horizontal position.