GWM 1.7

What is the definition of an even function

A function f(x) is an even function if f(-x) = f(x) for all values of x

What is the definition of an odd function?

A function g(x) is an odd function if g(-x) = -g(x) for all values of x

True or false: all functions are either even or odd

False

How can you graph the function y = ah(x) + k, where a and c are constants and h(x) can be expressed as a composite function f(g(x)), where g(x) = (x - h)?

First, the "base graph" is the graph for f(x) Next, shift the "base graph" over h units to the right to form a new graph (call it "new graph 1") (note -- if g(x) in the composition is x + h, shift the "base graph" over h units to the left to form "new graph 1" Next, if a < 0, flip the graph around the x-axis to form "new graph 2"; you then "stretch"/"contract" the graph by multiplying all y-values by |a| Finally, shift the graph up by "k" units if k > 0; shift the graph down by "k" units if k < 0

What is the rule of thumb for telling whether a polynomial is an even function

If all of the terms either have an even degree or is a constant, then the function is an even function

What is the rule of thumb for telling whether a polynomial is an odd function?

If all of the terms have an odd degree (an exponent of '1' counts as odd)

What is characteristic of the graph of an even function?

The graph of an even function is symmetric about the y-axis

What is characteristic of the graph of an odd function?

The graph of an odd function is symmetric about the origin, i.e. if the point (x,y) lies on the graph of an odd function, so will the point (-x,-y)

When you have a rational function f(x), how can you tell what values of x may give holes and/or vertical asymptotes?

The values of x where the denominator of f(x) is equal to 0 are the values that will give either vertical asymptotes or holes

If you know that a function f(x) has a hole at x = c, how do you find the y-value of the hole?

The y-value of the hole will be at f*(c), where f*(x) is the function after factoring and canceling

Given that you know that a value of x gives either a vertical asymptote or a hole for a function f(x), how can you tell which it is?

You look at f*(x), the function after factoring and canceling If f*(x) is undefined at that value of x, then that value of x gives a vertical asymptote If f*(x) is defined at that value of x, then that value of x gives a hole

How can you find the horizontal asymptotes of a rational function?

You look at the dominant term of the numerator and denominator and compare their degrees: (1) If the dominant term of the numerator has a smaller degree than the dominant term of the denominator, then the horizontal asymptote is y = 0 (2) If the dominant term of the numerator has a degree equal to the dominant term of the denominator, then the horizontal asymptote is y = (ratio of the dominant terms) (3) If the dominant term of the numerator has a greater degree than the dominant term of the denominator, then there is no horizontal asymptote