MAT 117 WEEK 4 LESSON PLAN
Total estimated time : 145 minutes
Objectives:
1.
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Graphs of Functions
Find the domain and range using the graph of a function
Vertical Line Test
Describe the increasing and decreasing behavior of a function
Classify a function as even or odd
Identify six common graphs
2. Transformations of Functions
 Sketch the graph of a function using common graphs and transformations
 Write the equation of function using common graphs and transformations
Motivation: [2 minutes]
Why learn about the graphs of functions? One reason is because it helps us recognize the
type of relationship that exists between two quantities being considered. Additionally, if
we’re analyzing a set of data, we’re able to use our knowledge of the different types of
graphs of functions to choose the one that best fits the pattern of the data being studied.
This allows us to then express the relationship between the two quantities being studied
in terms of an equation, which we can then use to estimate, predict, and describe all sorts
of things concerning the situation being studied. Individual teachers may want to relate a
personal experience with graphs, data, etc.
Warm up discussion: [5 minutes]
Provide an example of an equation for a function and draw the graph. Prompt students to
point out properties of the graph previously learned such as intercepts and symmetry.
1. Graphs of functions
Find the Domain and Range using the graph of a function
Warm up example or activity:
[ 5 minutes] Prompt students to review meaning of domain and range using input/output,
independent/dependent variable, and horizontal/vertical axis ideas. On board or
overhead, provide graph for students to determine what values constitute the domain and
range. Try to stay away from using x and y values because functions are not always in
terms of those variables.
Formal concept:
[ 3 minutes] State the definitions combined with the concept of the graph: the domain is
all possible input values on the horizontal axis that give a defined output, which are the
range values on the vertical axis.
Example:
[ 10 minutes] Provide several different graphs and prompt students to find the domain
and range. Review the different ways to express sets of numbers, such as set-builder
notation, interval notation, etc, and indicate your preference if you have one. Be sure to
use examples that include infinite and finite domain/ranges and the corresponding
different notation.
Some good ones to start with:
f ( x)   x 2  8 x  16
g ( x)  x  2
A step function, such as the greatest integer function.
Vertical Line Test
Warm up example or activity:
[ 10 minutes] Draw a function diagram on the board and ask the students how to connect
inputs and outputs so that it does not represent a function. (Hopefully you will have one
input in the domain going to at least two outputs in the range.) Show them what this
means in terms of two coordinate pairs and plot these on a set of axes. Point out that this
wouldn’t represent a function because one point is right above the other. Some students
will catch on a say vertical line test, if not, point out the concept to them. Remind them of
the definition of a function, the output must be unique, or there must be at most one
output value for each input.
Formal concept:
[ 2 minutes] Write a formal definition of the vertical line test on the board.
Example:
[ 6 minutes] Draw several graphs on the board and apply the VLT. Include tricky
examples, like a graph of a step function or one consisting of plotted points.
Increasing and decreasing behavior of a function
Warm up example or activity:
[ 5 minutes] Introduce a function that represents distance D (in miles) from a certain
point as a function of time t (in hours). For example, D(t) = 6t – t² for 0 ≤ t ≤ 6. Prompt
students to sketch the graph of the function. Ask students to interpret the graph within
some context. Prompt students to describe where the function is increasing, decreasing,
or neither within same context. Remind students that a function’s graph is very useful in
determining a function’s behavior. This example will hopefully show the students that
there is a purpose in looking at this type of behavior.
Formal concept:
[ 3 minutes] Explain that increasing and decreasing behavior must be expressed in terms
of the input variable. Also remind the students to look at the graph from left to right
along the horizontal axis.
Example:
[ 10 minutes] Do a straightforward example (not a word problem) that has more turning
points. For instance, f(x) = 2x³ – 6x² – x + 5. After this initial example, a word problem
may be appropriate.
Also, introduce a graph that is constant for a small portion of the domain.
y
x
Even and odd functions
Warm up example or activity:
[ 6 minutes] Review y-axis, x-axis, and origin symmetry of graphs, including a discussion
of the relationship between the points. For example, if a graph is symmetric about the yaxis and includes the point (3,7), then the point (–3,7) must also be on the graph. Prompt
students to recall the algebraic method of determining whether a graph has y-axis
symmetry, substituting -x for x in the graph’s equation, etc., and then make the
connection to functions using f(x). Similarly, discuss origin symmetry. A discussion of
why we don’t consider x-axis symmetry when dealing with functions might be a good
review of the definition of function, VLT, etc.
Formal concept:
[ 2 minutes] State the formal definitions of even and odd that were developed above.
Examples:
[ 10 minutes] Example 1: Draw several graphs and decide if they are even, odd, or
neither.
Example 2: Given functions, algebraically determine if they are even, odd, or neither.
Ask the students to graph the functions to check and see if their answers are consistent. A
good one to start with:
x
f ( x)  2
(you can look at graphs of rationals on the calculator to clear up future
x 1
questions. )
Identify six common graphs
Warm up example or activity:
[ 12 minutes] Sketch the graphs prompt students to identify their corresponding
functions. Stress the importance of recognizing the general behavior of these graphs and
how one is distinguished from another. Have the students state properties previously
learned, like increasing/decreasing and even/odd functions as well as domain and range
of each.
2.Transformations of functions
Vertical and horizontal shifts
Warm up example or activity:
[10 minutes] Let the students work in groups and explore functions such as
g(x)=(x+4)² and f(x)=|x| – 3 on their calculators. Ask them to write out any rules they
find.
Formal concept:
[ 3 minutes] Define the different types of shifts algebraically and clearly state how they
need to write them out in words. For example, f(x) + c means a vertical shift up c units
for some c > 0.
Example:
[ 5 minutes] Combine various shifts into the same problem and have the students state all
transformations. Then sketch the graph in terms of the common graph for each.
f(x)=(x – 2)³ + 5.
Vertical and horizontal stretches and compressions
Warm up example or activity:
[ 10 minutes] Again, let the students work in groups and explore various functions like
f(x) = 4x² and g(x) = (3x)².
Formal concept:
[ 3 minutes] Define the different types of stretches and compressions algebraically and
clearly state how they need to write them out in words. For example, c•f(x) means a
vertical stretch by a factor of c units for some c > 1 and a vertical compression for
0 < c < 1.
Example:
[ 5 minutes] Combine stretches and compressions with shifts, state all transformations,
and then graph in terms of the common graph. Use examples such as h(x)= 2 x  4  5
or g(x) = ½ x³ + 4.
Reflections about the x and y axes
Warm up example or activity:
[ 10 minutes] With the students working in groups, let them explore reflections. Include
an even functions so they will see what happens with a reflection about the
y-axis. Ask the students why we don’t talk about reflections about the x-axis and make a
connection to even and odd functions.
Formal concept:
[ 3 minutes] Define the difference between the two types of reflections. For example,
f(–x) will cause the graph of f(x) to reflect about the y-axis.
Example:
[ 5 minutes] Combine reflections with stretches, compressions and shifts, state all
transformations, and graph in terms of the common graph. Be sure to mention that the
order the transformations are applied to the original graph can make a difference. Use
examples such as
f(x)= –2 | x –1 | +3 .
Follow up assessment
Homework is sufficient.