FUNCTIONS
Introduction to Functions
Overview of Objectives, students should be able to:
1.
2.
3.
4.
5.
Find the domain and range of a relation
Determine whether a relation is a function
Evaluate a function
Recognize functions in tables
Use functions as models and make future
predictions.
Main Overarching Questions:
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•
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How do you determine the domain and range of a relation?
Compare/contrast relations and functions.
How can you determine whether a relation is a function from a table of values?
Objectives:
Activities and Questions to ask students:
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Find the domain and range of a relation
1.
•
•
Determine whether a relation is a function
Recognize functions in tables
2.
1.
2.
•
Evaluate a function
1.
2.
•
Use functions as models and make future
1.
Ask students if they remember what the domain and range of a function represent. (x and yvalues, input/output, independent/dependent variables).
Provide a list of ordered pairs, and have students identify the domain and range.
Ask students what the word function means in every day language. Solicit several
definitions/descriptions. Building upon student responses, ask, “Would it be correct to say that
my paycheck is a function of the amount of time that I work, or that the amount of time that I
work is a function of my paycheck?” (Paycheck is a function of time) Relate this to x and y values
(time x, paycheck y). If I get paid $8 per hour, what equation would represent my pay?
Show students several examples of x/y charts, some representing functions and some not
representing functions. Ask students to work in pairs to determine which charts fit the definition
of a function. Discuss the results as a class.
Using the equation from the paycheck example above, have students evaluate the pay for
several different inputs. Ask for general observations about the x- and y-values (ex. The pay
depends on the number of hours worked).
In a function, the y value acts as a function of the x value. The y-value changes as a result of
changes to the x-value, just like my paycheck increases or decreases depending upon how many
hours I work. Just as each time produces a unique pay, each x-value produces a unique y-value in
a function. In other words, for every x-value, there is exactly one y-value.
Provide a quadratic equation that models the height of a trajectory over time. Ask students to
evaluate the function for several different x-values (sufficient to see repeating y-values). Ask if
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.
predictions.
this equation would represent a function.
2. Have the students use the quadratic model to predict the height of the trajectory at specific
points in time (1 sec, 2 sec, 3 sec, etc.). Discuss the significance of using equations to model
functions and how they can be used to make predictions without continuing the pattern forever.
3. Provide a problem set for independent practice.
Graphs of Functions
Overview of Objectives, students should be able to:
1. Graph functions by plotting points or through a table
2. Use the vertical line test to identify functions
3. Obtain information about a function from its graph
a. Find f(c), where c is a constant
b. Find x in f(x) = c, where c is a constant.
4. Identify domain and range from a function’s graph.
5. Recognize the connection between finding zeros,
roots, x-intercepts, and solutions to a function.
Objectives:
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Graph functions by plotting points or through a table
•
Use the vertical line test to identify functions
Main Overarching Questions:
•
•
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How can you determine whether a relation represents a function from a graph?
How can you identify the domain and range of a function from the graph?
What does it mean to “find the zeros” of a function?
Activities and Questions to ask students:
1. Using the paycheck example from the previous section, have students recreate a table of
values and use those points to plot a line graph. (You may want to discuss the fact that this
graph would be a ray, rather than a line, since time can’t be negative.)
2. Ask students what the points on the graph represent. (Many students have a hard time
making the connection between the solutions to an equation and points on the graph. Make
sure to emphasize that x values are inputs to the function and y values are outputs).
**Possible graphing calculator usage:
Have students input the function and graph using the graphing calculator. Show students how
to generate a table of values using the calculator.
1. Using the tables of values from the previous section, have students plot the points. Ask them
to find similarities between the graphs of functions and the graphs of non-functions.
2. Show students how the vertical line test can be used to quickly determine whether or not a
graph is a function. Provide several examples and non-examples in addition to the studentgenerated graphs (circles, parabolas, horizontal and vertical lines, etc.).
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
Obtain information about a function from its graph
I. Find f(c), where c is a constant
II. Find x in f(x) = c, where c is a constant.
•
Identify domain and range from a function’s graph.
•
Recognize the connection between finding zeros,
roots, x-intercepts, and solutions to a function.
3. Provide a brief problem set for small group or independent practice.
**Possible graphing calculator usage:
Introduce the “Stat Plot” feature on the graphing calculator, and have students input the
values from the x/y charts and generate a graph. Have students use the vertical line test to
determine whether or not each set of values represents a function.
1. Using the line graph from the paycheck example, ask students if they can determine how
much you would make if you worked x number of hours. (Example: How much would I make
if I worked 7 hours?). Encourage students to use the graph only, rather than the equation. Be
sure to use function notation ( f(7) = ___).
2. Using the line graph, ask students to determine how long it would take to earn y dollars.
(Example: How long would I need to work in order to earn $80?). Again, encourage students
to draw conclusions from the graph, rather than the equation. Be sure to use function
notation ( f (___) = 80).
3. Provide a problem set for independent practice.
**Possible graphing calculator usage:
Have students re-enter the paycheck function, and show them how to evaluate the function
for a specific x-value using the “calculate: value” feature. Emphasize that when using this
feature, the calculator marks that point on the graph.
1. Review the definitions of domain and range. Provide a table of values, and ask students to
identify the domain and range. Have students plot the points from the table, and then ask
how they would determine the domain and range if they only had the graph.
2. Provide a few examples of continuous graphs, and ask students to work in small groups or
pairs to discuss ways to identify the domain and range of each graph. Have students share
solutions and methods with the whole group.
o Using the graphs from the previous exercise, ask students to identify the x-intercepts for
each graph. What is the value of y at each of those points?
o Recognizing that y = 0 at each x-intercept, how could we identify the x-intercepts of a
function given its equation only? Provide several examples of function equations and have
students work in small groups or pairs to determine the x-intercepts for each.
o Explain to students that finding the x-intercepts is also called “finding the zeros” of a
function, and that this terminology will be used throughout the rest of the course.
o Provide a problem set for independent practice.
Possible graphing calculator usage:
Provide several additional function equations, and allow students to graph each equation
using the calculator. Introduce the “calculate: zero” feature.
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.
The Algebra of Functions
Overview of Objectives, students should be able to:
1. Find the domain of the function in equation form
a. Find the domain of a linear, quadratic,
polynomial, rational, and radical function.
b. Find the domain of a square root function
c. Find the domain of a rational function
2. Use the algebra of functions to combine functions
a. Find the sum function and its domain
b. Find the difference function and its domain
c. Find the product function and its domain
d. Find the quotient function and its domain
Main Overarching Questions:
i.
i.
What could cause the domain of a function to be restricted?
When you add, subtract, multiply, or divide functions, how is the domain affected?
Objectives:
Activities and Questions to ask students:
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
Find the domain of the function in equation form
o Find the domain of a linear, quadratic,
polynomial, rational, and radical function.
o Find the domain of a square root function
o Find the domain of a rational function
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Ask students to recall how to determine the domain and range of a relation (look at the x/y
values on the table, look at the x/y values from the points on the graph). Because functions are
special types of relations, every function also has a domain and a range.
Provide a linear function, and then ask students to discuss in pairs how they would determine
the domain of that function. Ask individuals to share out responses after a few minutes. (Create
a t-chart, plot points, etc.)
Lead the class in creating a short table of values for the linear function. Ask students, “Would
this table represent all of the possible x and y values for this function? How many different
numbers could I substitute for x? How many possible output values are there for y?” Guide
students toward understanding that you could plug in an unlimited number of x-values, so the
domain would be unrestricted.
Provide the students with the equation for a horizontal line (ex. f(x) = 5). What is the domain for
this function? What is the range?
Provide a simple quadratic function for students (ex. f(x) = x^2 + 5). Ask students to work in pairs
again to determine a method for finding the domain. Have students share their methods with
the entire group.
Guide students through a similar process as the linear function, creating a table of values and/or
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.



graphing. Is there any x-value that would be invalid as an input for this function? Would this be
the case for any quadratic function?
Provide a similar example and use a similar process for a polynomial function.
Provide an example of a basic rational function (ex. f(x) = 4/(x-2). Ask students to work in pairs to
determine the domain of the function. Ideally, students will recognize a problem when they try
to substitute x = 2. Ask students why the error occurred. Can x = 2 in this function? Why not?
Discuss the implications for the domains of rational functions. Guide students to the conclusion
that any input value that causes the denominator to equal zero should be excluded from the
domain of a rational function. Provide several additional examples from whole group or small
group practice.
Provide an example of a basic radical function (ex. f ( =
x)
x − 1 ). Have students work in pairs
to determine the domain of the function. Ideally, students will recognize a problem when they
try to input values that are less than 1. Ask students why the error occurred.
 Discuss the implications for the domains of radical functions. Guide students to the conclusion
that any input value that causes the radicand to be negative should be excluded from the
domain of a radical function. Provide several additional examples for whole group or small group
practice.
 Briefly review the situations that can create restrictions on the domain of a function (zero in the
denominator, negative radicand). Provide a problem set for independent practice.
**Possible graphing calculator usage:
After students have identified the domains for each of the sample functions above, allow them to
graph them in the calculator. Assist them in making connections between the domains and the
graphs.
1. Review the terms sum, difference, product and quotient. Solicit student definitions for each.
2. Sometimes it is useful to combine two or more functions into a single function. Given that
f(x) = 2x + 1 and g(x) = -3x -4, how could we simplify f(x) + g(x)? Have students work in pairs
to come up with an answer. Ask individuals to share methods with the entire group.
3. What were the domains of the original functions f(x) and g(x)? What is the domain of f(x) +
g(x)?
4. Using the same functions, ask pairs to determine the difference and share methods.
Emphasize that you are subtracting the entire value of g(x), showing students how to
distribute the subtraction sign to each term.
5. Discuss the domain of the difference. Is it any different than the domain of the sum?
6. Using the same functions, ask pairs to determine the product and share their methods with

• Use the algebra of functions to combine functions
o Find the sum function and its domain
o Find the difference function and its domain
o Find the product function and its domain
o Find the quotient function and its domain
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.
7.
8.
9.
10.
the whole group. Emphasize the fact that each term in the first function must be multiplied
by each term in the second function by distributing (you may need to review FOIL and
distributive property).
Discuss the domain of the product. Has it changed from the original functions and if so, how?
Repeat the process for the quotient of the functions. Discuss changes to the domain,
emphasizing that any value that makes the denominator zero should be excluded from the
domain.
Model additional examples using quadratic, polynomial, radical, and rational functions. For
each, ask students to determine the domain of the original functions and the domain of the
resulting function. Re-emphasize that denominators of zero and negative radicands will
result in restrictions to the domain.
Provide a problem set for independent practice.
Quadratic Functions and their Graphs
Overview of Objectives, students should be able to:
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•
•
•
Find the intercepts
a. x-intercepts by solving a quadratic
equation
b. y-intercept
Find the vertex of a quadratic function
a. Use the formula to find the vertex
b. Determine if the vertex is a maximum or
minimum point
Determine if a quadratic function’s graph opens
up or down
Recognize the connection between finding zeros,
roots, x-intercepts, and solutions to a quadratic
function.
Objectives:
Main Overarching Questions:
• How can you find the x-intercepts of a quadratic function on the graph and from the
equation?
• How can you find the y-intercepts of a quadratic function from the graph and from
the equation?
• How can you find the vertex of a quadratic function from the graph and from the
equation?
• How can you determine whether the vertex of a quadratic function represents a
minimum or a maximum value?
• How can you tell whether the graph of a quadratic function should open up or down
given its equation?
•
Explain how to “find the zeros” of a quadratic function.
Activities and Questions to ask students:
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
Find the intercepts
o x-intercepts by solving a quadratic equation
o y-intercept
o by looking at graph
•
Find the vertex of a quadratic function
o Use the formula to find the vertex
o Determine if the vertex is a maximum or
minimum point
1. Draw a parabola on the board (or provide a graphed parabola) and ask students to identify
the y-intercept. When the graph crosses the y-axis, what is the value of x?
2. Provide a quadratic function, and ask students to find the y-intercept (plug in zero for x).
Provide a few additional examples for independent or small group practice. After students
have found solutions, ask them if there is a shortcut for determining the y-intercept of a
quadratic function (just use the “c” value).
3. Using the original parabola, ask students to identify the x-intercepts. When the graph crosses
the x-axis, what is the value of y?
4. Provide a quadratic function, and ask students to find the x-intercepts (plug in zero for y, or
f(x) ). Ask students to recall methods for solving quadratic equations from previous lessons
(factoring, completing the square, quadratic formula). Which method would be most
efficient for this problem?
5. Emphasize the idea that finding the x-intercepts of a quadratic function is the same as solving
a quadratic equation. This is also called “finding the zeros,” “finding the roots,” or “finding
the solutions” of a quadratic function.
6. Guide students through several additional examples, asking them to determine which
method to use for each. Provide a problem set for independent practice.
**Possible graphing calculator usage:
After students have identified intercepts using the equations, allow them to graph the
functions and check their answers using the “calculate: value” and “calculate: zero” features.
1. Quadratic functions can be used to model many different real-world problems. One of the
most common is the path of a trajectory. For example, the height of a punted football could
be modeled with the quadratic function f ( x) =
−0.01x 2 + 1.18 x + 2 , where the horizontal
distance in feet from the kicker’s foot is x, and f(x) represents the height of the ball in feet.
2. If I were to graph this function, where would the y-intercept be? What about the xintercepts? Allow students to work in pairs to find x- and y-intercepts. What do the
intercepts represent?
3. Plot the x- and y-intercepts on a graph, and then ask students to predict where the vertex
would be located. (You may need to review that the vertex is the highest or lowest point on
the graph). Guide students to the conclusion that the vertex would be exactly halfway
between the two x-intercepts (at x-value 59).
4. Would the vertex be located above or below the x-intercepts? Assist students in
understanding that because the ball is being kicked upward, the vertex would be above the
x-intercepts.
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
Determine if a quadratic function’s graph opens up
or down
5. How can we tell how high the ball actually went? Solicit student responses, guiding students
to the conclusion that the height can be found by plugging in the corresponding x-value (59
feet). (Highest point, 36.81 feet).
6. Provide students with the formula y = -b/2a. Ask them to use the ‘a’ and ‘b’ values from the
football example to solve the formula. What do they notice? (The result is the same as the xvalue of the vertex). Clarify that this is the formula for finding the x-value of the vertex of any
quadratic function, and that it is usually faster to use this formula than to determine the
midpoint of the x-intercepts.
7. In the football example, would you say that the vertex was the maximum or minimum of the
function? From the graph, it is relatively easy to recognize that it is a maximum because it is
the highest point on the graph. Is there a way to tell whether the vertex is a maximum or
minimum without actually graphing? Solicit student responses, guiding them to the
conclusion that (-a) results in a downward facing graph with a maximum value at its vertex,
and that (a) results in an upward facing graph with a maximum value at its vertex.
8. Provide a problem set for small group or independent practice, asking students to identify
the vertex, the direction of the graph, and whether the vertex represents a maximum or a
minimum.
**Possible graphing calculator usage:
Allow students to graph the football equation in the calculator. Introduce the “calculate:
minimum/maximum” feature and allow students to verify their vertices using the calculator.
• Covered in previous section.
**Possible graphing calculator usage:
Provide several equations for quadratic functions (some with positive a values, and some with
negative). Allow students to graph the functions in the calculator, and ask them to draw a
conclusion about the relationship between the equation and the direction of the graph.
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.