College Algebra Lecture Notes Section 2.5 Page 2 of 2

Section 2.5: Analyzing the Graph of a Function

Big Idea: .

Big Skill: .

A. Graphs and Symmetry

Even Functions: y-Axis Symmetry

A function f is an even function if and only if each point (x, y) on the graph of f, the point (-x, y) is also on the graph. In function notation, . The graph of an even function looks like a mirror image if a mirror is placed on the y-axis.

Even Functions: Symmetry about the Origin

A function f is an odd function if and only if each point (x, y) on the graph of f, the point (-x, -y) is also on the graph. In function notation, . A line through the origin intersects the graph of an odd function at points equidistant from the origin.

Practice:

  1. .
  2. .

B. Intervals Where a Function is Positive or Negative

·  … are separated by zeroes of the function

Practice:

  1. .

C. Intervals Where a Function is Increasing or Decreasing

·  … are separated by maxima or minima of the function.

Increasing and Decreasing Functions

Given an interval I that is a subset of the domain, and , and :

·  A function is increasing on I if for all .

·  A function is decreasing on I if for all .

·  A function is constant on I if for all .

Practice:

  1. .

D. More on Maximum and Minimum Values

·  A function f has a local maximum at x = a if for all x-values in the neighborhood of a.

·  A function f has a global (or absolute) maximum at x = a if for all x-values in the domain.

·  A function f has a local minimum at x = a if for all x-values in the neighborhood of a.

·  A function f has a global (or absolute) minimum at x = a if for all x-values in the domain.

·  If the domain of a function is restricted to a closed interval, then there can also be an endpoint maximum or endpoint minimum at each endpoint.

·  A quadratic function has an absolute extremum at its vertex.

Practice:

  1. .

E. Rates of Change and the Difference Quotient

Average Rate of Change

For a function that is smooth and continuous on an interval containing x1 and x2, the average rate of change between x1 and x2 is given by

(provided ).

This is also the slope of the secant line through.

The Difference Quotient

For a function f(x) that is smooth and continuous and a constant h ¹ 0 on an interval containing x and x + h, the difference quotient is given by

.

This is also the slope of the secant line through.