MATH 1142
Section 2.1 Worksheet
NAME
Describing Graphs of Functions
We have already briefly discussed some characteristics of functions and their graphs in chapter 0. In this chapter,
we will first discuss how the derivative of a function can provide information about the function itself. Before we
do this, lets look more closely at the aforementioned characteristics.
Increasing and Decreasing
We have already informally discussed these concepts–let’s not give more formal definitions.
We say a function f (x) is increasing on an interval if whenever x1 and x2 are on the interval withx1 < x2 , we
have f (x1 ) < f (x2 ).
Sketch a picture of what this means.
What does it mean for a function to be decreasing on an interval?
One important thing to note is that while a function may be increasing on an interval, it does not mean that its
rate of change is constant.
Also, notice that we talk about functions as being increasing and decreasing on intervals. We can also refer to
functions as being increasing and decreasing at a point, x = c. In this case, what we mean is that the function is
increasing or decreasing on an interval containing c.
We talking about functions being increasing or decreasing on intervals because many functions are not increasing
or decreasing on their entire domain but can fluctuate between the two. When a function changes from increasing
to decreasing or vice versa it creates a an extremum.
Extreme Points
Sketch a picture of what happens when a function changes from
increasing to decreasing
and
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decreasing to increasing.
A relative maximum is when a graph changes from
to
.
A relative minimum is when a graph changes from
to
.
We us the adjective relative to indicate that a given graph may have more than one. Sometimes the word local is
used instead. The absolute maximum value of a function is the largest value that the function assumes on the
domain and the absolute minimum value is the smallest.
Sketch a picture of the tangent line to the curve at a
maximum
and
minimum.
Concavity
We say a function is concave up at x = a if there is an open interval containing a throughout which the graph
of the function lies above it’s tangent line. A function is concave down at x = a if there is an open interval
containing a throughout which the graph of the function lies below it’s tangent line.
Sketch picture described by these definitions.
When a function has a relative maxima or minima there is a way of describing what is happening around that
point. Around a maximum, the function is concave down and around a minimum the function is concave up.
Notice that there is a point on a continuous function where it changes from being concave up to concave down
and vice versa. This point is called an inflection point.
Intercepts, Undefined Points, and Asymptotes
Intercepts, undefined points and asymptotes are the last few things that we would like to be able to identify when
describing the graph of a function.
As we have seen with rational functions, vertical asymptotes can be identified by looking at where a rational
functions are undefined. Horizontal asymptotes can be found by looking at the end behavior of a function i.e.
limx→∞ and limx→−∞ .
In summary these are the six things to pay attention to when describing a graph:
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Describing a Graph
1. intervals of increasing or decreasing and relative maximum and minimum
points
2. absolute maximum and minimum values
3. intervals in which the function is concave up or down and inflection points
4. intercepts
5. undefined points
6. asymptotes
Examples: Describe each of the following graphs in terms of the above six categories.
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