Section 1.3 A Whole Lotta Stuff

Increasing, Decreasing, and Constant The graph of an increasing function goes uphill from left to right. The graph of a decreasing function goes from left to right. The graph of a constant function is level from left to right. These behaviors occur on subintervals of the domain.

Use the graph below to determine increasing, decreasing, and constant intervals. Use interval notation.

Relative Extrema The points at which a graph changes from increasing to decreasing or decreasing to increasing can be used to find relative minima (think: valley) or relative maxima (think: mountain). The x-coordinate of the point is the location. The y-coordinate of the point is the value.

Identify the Relative Extrema

Even and Odd Functions Even functions are symmetric about the y-axis. If a function has all even exponents, it is an even function. Odd functions are symmetric about the origin. If a function has all odd exponents, it is an odd function. If you cannot tell by looking at exponents, graph the function and look for symmetry.

Even, Odd, or Neither?

Piecewise Functions A piecewise function is a function made up of more than one function rule, each with its own domain set. Which function you “plug” into is determined by the input value and to which domain set it belongs. A “real world” example of a piecewise function is standing in line according to the first letter of your last name.

Evaluating a Piecewise Function Determine which interval contains your input value Substitute into the function rule that corresponds to that interval Do not substitute into more than one function!

Graphing a Piecewise Function Graph each function on the same set of coordinate axes. Use the conditions on each function to create a table of values. Be sure to note which endpoint needs to be “open” and which endpoint needs to be “closed”. In finding the range, recall that range is “ smallest value of y to largest value of y.”

A Rather Large Example Find f(-2) Find f(1) Find f(4) Sketch the graph. Find the range of the function. Use interval notation.

The Difference Quotient The difference quotient is used in calculus. Calculating the difference quotient is an application of evaluating a function. Be very careful when finding f(x + h). (x + h)2 = x2 + 2xh + h2, not x2 + h2

Find the Difference Quotient for each of the following: