Download Report

Problem 1
Students evaluate absolute
value expressions. They will
complete a table of values for
an absolute value equation, and
then graph the table of values.
Students conclude the graph
is a function but not a linear
function.
Problem 1
The V
Recall that the absolute value of a number is defined as the distance from the number to
zero on a number line. The symbol for absolute value is | x |.
1. Evaluate each expression shown.
a. | 23 | 5 3
b. | 11 | 5 11
2 5 5__
2
c. 25__
3
3
d. | 110.89 | 5 110.89
|
Grouping
|
2. Use the function y 5 | x |, to complete the table.
Have students complete
Questions 1 through 8 with
a partner. Then share the
responses as a class.
Share Phase,
Questions 1 and 2
t What is the distance from -3
to zero on a number line?
t What is the distance from 11
-JMXMWWXEXIH
XLEX]SYEVI[SVOMRK
[MXLEJYRGXMSR[LEX
HSIWXLEXXIPP]SYEFSYX
XLIVIPEXMSRWLMTFIX[IIR
XLIMRTYXERHSYXTYX
ZEPYIW#
to zero on a number line?
t What is the distance from
2 to zero on a number
-5 __
3
line?
t What is the distance from
x
y 5 |x|
27
7
23
3
21
1
20.5
0.5
0
0
2
2
4
4
7
7
E
108

Chapter 2
Linear Functions
© 2011 Carnegie Learning
110.89 to zero on a number
line?
Share Phase,
Questions 3 through 8
t How would you describe the
3. Graph the values from the table on the coordinate plane.
graph of this absolute value
equation?
y
8
t Is the graph considered a
6
4
function? Why or why not?
2
t Does the graph pass the
x
–8
–6
–4
–2
vertical line test? Explain.
t What is the maximum or
function different than a
linear function?
t How is an absolute value
function similar to a linear
function?
4
6
8
–4
–6
greatest value of y? How do
you know?
t How is an absolute value
2
–2
–8
4. Connect the points to model the relationship of the equation y 5 | x |.
See graph.
5. What is the domain of this function? Do all the points on the graph
make sense in terms of the equation y 5 | x |. Explain your reasoning.
/IITMRQMRH
XLIHSQEMR
ERHVERKI
VITVIWIRXWIXW
SJRYQFIVW
The domain is the set of all numbers. It makes sense to connect these points
because the absolute value can be determined for every number.
6. Does the graph of these points form a straight line? Explain your reasoning.
No. The points go down and then back up. The distance from the number to
zero gets smaller as the points get closer to zero and then larger as the points
get farther away.
7. What is the minimum, or least value of y? How do you know? State the range of
this function.
The least value of y is zero. The range is all numbers greater than or equal to 0.
8. Is this a linear function? Explain your reasoning.
No. This is not a linear function because its graph is not a straight line, but is more
like two parts of two different lines.
© 2011 Carnegie Learning
You have just graphed an absolute value function. An absolute value function is a
function that can be written in the form f(x) 5 | x |, where x is any number. Function notation
can be used to write functions such that the dependent variable is replaced with the name
of the function, such as f(x).
E
2.7
Introduction to Non-Linear Functions

109
Problem 2
Students calculate the area of
squares, given the length of
a side. They will complete a
table of values for a quadratic
equation, and then graph
the table of values. Students
conclude the graph is a function
but not a linear function and not
an absolute value function.
Problem 2
Not V but U
Recall that the area of a square is equal to the side length, s, multiplied by itself and is
written as A 5 s2.
1. Calculate the area of squares with side lengths that are:
a. 3 inches.
A 5 s2 5 (3)2 5 9 square inches
b. 5 feet.
A 5 s2 5 (5)2 5 25 square feet
Grouping
c. 2.4 centimeters.
Have students complete
Questions 1 through 8 with
a partner. Then share the
responses as a class.
A 5 s2 5 (2.4)2 5 5.76 cm2
5 inches.
d. 12__
8
10,201
5 2 5 ____
101 ____
101 5 _______
25 square inches
5 159___
A 5 s2 5 12__
64
8
8
8
64
( ) ( )( )
In the equation A 5 s2 the side length of a square, s, is the independent variable and the area
of a square, A, is the dependent variable. This formula can also be modeled by the equation
Share Phase,
Questions 1 and 2
t What unit of measure is used
y 5 x2, where x represents the side length of a square and y represents the area of a square.
2. Use the equation, y 5 x2, to complete the table.
x
y 5 x2
t What unit of measure is used
23
9
22
4
21
1
20.5
0.25
0
0
2
4
2.3
5.29
3
9
to describe the area of a
square?
t What is the sign of a negative
number after it has been
squared?
t In the table of values, is
it possible for y to have a
negative value? Why or why
not?
E
110

Chapter 2
Linear Functions
(SIWXLMW
IUYEXMSR
VITVIWIRXE
JYRGXMSR#
© 2011 Carnegie Learning
to describe the length of a
side of a square?
Share Phase,
Questions 3 through 8
t How would you describe
3. Graph the values from the table on the coordinate plane.
y
the graph of this squared
equation?
8
6
t Is the graph considered a
4
function? Why or why not?
2
t Does the graph pass the
x
–8
function different than a
linear function?
t How is this quadratic
function similar to a linear
function?
t How is this quadratic
function different than an
absolute value function?
t How is this quadratic
–2
2
4
6
8
–4
t What is the maximum or
t How is this quadratic
–4
–2
vertical line test? Explain.
greatest value of y ? How do
you know?
–6
–6
–8
4. Connect the points to model the relationship of the equation y 5 x2.
See graph.
5. What is the domain of this function? Do all the points on the graph make sense in
terms of the equation y 5 x2. Explain your reasoning.
The domain is the set of all numbers. It makes sense to connect the points since
the square of a number can be determined for every number.
6. What is the minimum, or least value of y? How do you know? State the range of
this function.
The minimum value of y is zero since 0 3 0 5 0 and the product of any other
number and itself is greater than zero. The range is the set of all numbers greater
than or equal to 0.
function similar to an
absolute value function?
7. Does the graph of these points form a straight line? Explain your reasoning.
No. The points go down and then back up. Because the square of a number is
always positive, the points are in the first and second quadrants.
8. Is this a linear function? Explain your reasoning.
No. This is not a linear function because the graph is not a straight line but
© 2011 Carnegie Learning
looks like a U.
You have just graphed a quadratic function. A quadratic function is a function that can be
written in the form f(x) 5 ax2 1 bx 1 c, where a, b, and c are any numbers and a is not
equal to zero.
E
2.7
Introduction to Non-Linear Functions

111
Problem 3
Students calculate the volume
of cubes, given the edge length.
They will complete a table of
values for a cubic equation, and
then graph the table of values.
Students conclude the graph
is a function but not a linear
function, not an absolute value
function, and not a quadratic
function.
Problem 3
Not V or U
Recall that the volume of a cube is defined as the product of the length of one edge times
itself 3 times and is written as V 5 s3.
1. Calculate the volume of cubes with an edge length that is:
a. 2 inches.
V 5 s3 5 (2)3 5 8 cubic inches
b. 1.5 feet.
V 5 s3 5 (1.5)3 5 3.375 cubic feet
Grouping
c. 2.1 centimeters.
V 5 s3 5 (2.1)3 5 9.261 cubic centimeters
Have students complete
Questions 1 through 8 with
a partner. Then share the
responses as a class.
3 inches.
d. 1__
4
343
7 3 ____
23 cubic inches
3 3 5 __
5
5 5___
V 5 s3 5 1__
4
64
4
64
( ) ( )
In the equation V 5 s3, the side length of a cube, s, is the independent variable and the volume
of the cube, V is the dependent variable. This formula can also be modeled by the equation
Share Phase,
Questions 1 and 2
t What unit of measure is used
y 5 x3, where x represents the side length of a cube and y represents the volume of a cube.
to describe the edge length
of a cube?
x
y 5 x3
t What unit of measure is used
22
28
to describe the volume of
a cube?
21.5
23.375
21
21
20.5
20.125
0
0
1.5
3.375
2
8
2.1
9.261
t What is the sign of a negative
number after it has been
cubed?
t In the table of values, is
it possible for y to have a
negative value? Why or
why not?
E
112

Chapter 2
Linear Functions
(SIWXLMW
IUYEXMSR
VITVIWIRXE
JYRGXMSR#
© 2011 Carnegie Learning
2. Use the equation, y 5 x3, to complete the table.
Share Phase,
Questions 3 through 8
t How would you describe the
3. Graph the values from the table on the coordinate plane.
graph of this cubic equation?
y
t Is the graph considered a
8
function? Why or why not?
6
t Does the graph pass the
4
2
vertical line test? Explain.
x
t What is the maximum or
–8
similar to a linear function?
t How is this cubic function
different than an absolute
value function?
–2
2
4
6
8
–4
–6
–8
t How is this cubic function
t How is this cubic function
–4
–2
greatest value of y ? How do
you know?
different than a linear
function?
–6
4. Connect the points to model the relationship of the equation y 5 x3.
See graph.
5. What is the domain of this function? Do all the points on the graph make sense in
terms of the equation y 5 x3. Explain your reasoning.
The domain is the set of all numbers. It would make sense to connect the points
because the cube of a number can be determined for every number.
t How is this cubic function
similar to an absolute value
function?
t How is this cubic function
6. What is the minimum value of y? How do you know? State the range of this function.
There does not seem to be a smallest value since as x gets smaller the value of y
continues to get smaller. The range is the set of all numbers.
different than a quadratic
function?
t How is this cubic function
similar to an quadratic
function?
7. Does the graph of these points form a straight line? Explain your reasoning.
No. The points move upward, curve to the right, and move upward again.
8. Is this a linear function? Explain your reasoning.
© 2011 Carnegie Learning
No. It is not a linear function because the graph is not a straight line.
You have just graphed a cubic function. A cubic function is a function that can be
written in the form f(x) 5 a3x3 1 a2x2 1 a1x 1 a0.
E
2.7
Introduction to Non-Linear Functions

113
Talk the Talk
Given an equation, students
name and explain how it
represents a function.
Talk the Talk
You have just completed tables of values and graphs for three different
non-linear functions.
Grouping
Name each equation and explain how it represents a function.
Have students complete the
Talk the Talk with a partner.
Then share the responses as a
class.
●
y 5 |x|
Absolute value function
For each input value, there is one and only one output value.
●
y 5 x2
Quadratic function
For each input value, there is one and only one output value.
●
y 5 x3
Cubic function
For each input value, there is one and only one output value.
E
114

Chapter 2
Linear Functions
© 2011 Carnegie Learning
Be prepared to share your solutions and methods.