Find symmetry in, and transform, polynomial functions.

Assignment

  • All vocabulary copied into notes
  • p185 7–27 13–21

Review

Most of this section is review of section 1.2. Addition will shift your graph, while multiplication will scale it, with negatives flipping. Doing something to the entire function means you are effecting it vertically, while applying an operation to just the $x$ is for horizontal transformations (and usually in the opposite direction).

Even and Odd Functions

The one new concept here is even and odd functions, which both describe types of symmetry. Even functions are symmetrical about the $y$-axis (a horizontal flip), and odd functions are symmetrical about the origin (a rotation of 180°).

You can test for both of these algebraically. If a function is even, then $f(-x)=f(x)$. For example, if $f(x)=x^2$ and we want to test even symmetry, we plug in $-x$ and see if we get the same function. In this case we do since $f(-x)=(-x)^2=x^2$. Anything squared will just be the positive version.

The test for odd functions is $f(-x)=-f(x)$. Plugging in $-x$ will yield the original function, but negated. The simple example here is $f(x)=x^3$ since $f(-x)=(-x)^3 = -x^3$.