2.1 Describing Graphs of Functions

If we examine a typical graph the function y = f(x), we can observe that for an interval throughout which the function is defined, that the function might be increasing, decreasing or neither.

We say that a function is increasing on an interval if x1 and x2 are in the interval such that x1 < x2 and we have f(x1) < f(x2).

Further, we say that f(x) is increasing at x = c provided that f(x) is increasing in some open interval on the x-axis that contains c.

We say that a function is decreasing on an interval if x1 and x2 are in the interval such that x1 < x2 and we have f(x1) > f(x2).

Further, we say that f(x) is decreasing at x = c provided that f(x) is decreasing in some open interval on the x-axis that contains c.

Extreme Points

A relative extreme point ( relative maximum point or relative minimum point) of a function is a point at which its graph changes from increasing to decreasing or vice versa.

A relative maximum point is a point at which the graph changes from increasing to decreasing.

A relative minimum point is a point at which the graph changes from decreasing to increasing.

The maximum value of a function is the largest value that the function assumes on its domain.

The minimum value of a function is the smallest value that the function assumes on its domain.

Note: Functions might or might not have maximum and/or minimum values.

If a function has a maximum value or minimum value at the endpoint(s) of its domain, we say that the function has an endpoint extreme value.

Changing slope

Consider the next two graphs. Note that the graphs of both are increasing, but there is a difference in how they are increasing.

What is the difference?

Graph I

Graph II

We note that the slope of graph I is increasing while the slope of graph II is decreasing.

In application, we would say that the debt per capita depicted in graph I is rising at an increasing rate.

From graph II, we observe that the population is increasing at a declining rate.

Concavity

Concavity has a relationship to the tangent lines of a curve.

We say that a function is concave up at x = a if there is an open interval on the x-axis containing a throughout which the graph of f(x) lies above its tangent line.

We say that a function is concave down at x = a if there is an open interval on the x-axis containing a throughout which the graph of f(x) lies below its tangent line.

An inflection point is a point on the graph of a function at which the function is continuous and the concavity of the graph changes, i.e., goes from concave up to concave down, or concave down to concave up.

Use the terms defined earlier to describe the graph.

•For x < 3, f(x) is increasing and concave down.

•Relative maximum at the point x = 3.

•For 3 < x < 4, f(x) is decreasing and concave down.

•Inflection point at x = 4.

•For 4 < x < 5, f(x is decreasing and concave up.

•Relative minimum at x = 5.

•For x > 5, f(x) is increasing and concave up.

Intercepts, Undefined Points and Asymptotes

We have previously discussed the idea of intercepts.

Recall that

The x-intercept is a point at which a graph intersects the x-axis. (x,0)

The y-intercept is a point at which the graph intersects the y-axis. (0,y)

Note that a function can have at most one y-intercept. Otherwise, its graph would violate the vertical line test for a function.

A function may have 0 or more x-intercepts.

Recall that some functions are not defined for all values of x. For example,

xxf

1)( is not defined at x = 0

xxf )( is not defined for x < 0

Graphs sometimes straighten out and approach some straight line as x increases (or decreases).

Theses straight lines are called asymptotes.

Asymptotes of a graph may be horizontal, vertical or diagonal.

The horizontal asymptotes of a graph may be determined by calculating the limits

)(lim xfx

and )(lim xfx

If either limit exists, then the value of the limit determines a horizontal asymptote.

We often expect the graph of a function f(x), at a value x that would result in division by zero, to have a vertical asymptote.

We now have six categories for describing the graph of a function

1. Intervals in which the finction is increasing or decreasing, relative maximum/minimum points

2. Maximum/minimum values

3. Intervals in which a function is concave up or concave down, inflection points.

4. X-intercepts, y-intercepts

5. Undefined points.

6. Asymptotes