Transcript Interactive Chalkboard

Algebra 2 Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 2-1 Relations and Functions
Lesson 2-2 Linear Equations
Lesson 2-3 Slope
Lesson 2-4 Writing Linear Equations
Lesson 2-5 Modeling Real-World Data: Using
Scatter Plots
Lesson 2-6 Special Functions
Lesson 2-7 Graphing Inequalities
Example 1 Domain and Range
Example 2 Vertical Line Test
Example 3 Graph Is a Line
Example 4 Graph Is a Curve
Example 5 Evaluate a Function
State the domain and range
of the relation shown in the
graph. Is the relation a
function?
The relation is {(1, 2), (3, 3),
(0, –2), (–4, 0), (–3, 1)}.
Answer: The domain is
{–4, –3, 0, 1, 3}. The range
is {–2, 0, 1, 2, 3}. Each member
of the domain is paired with exactly one member of the
range, so this relation is a function.
State the domain and range
of the relation shown in the
graph. Is the relation a
function?
Answer: The domain is
{–3, 0, 2, 3}. The range
is {–2, –1, 0, 1}. Yes, the
relation is a function.
Transportation The table
shows the average fuel
efficiency in miles per
gallon for light trucks for
several years. Graph this
information and
determine whether it
represents a function.
Year
Fuel Efficiency
(mi/gal)
1995
20.5
1996
1997
1998
1999
20.8
20.6
20.9
20.5
2000
2001
20.5
20.4
Year
Fuel Efficiency
(mi/gal)
1995
20.5
1996
1997
1998
1999
20.8
20.6
20.9
20.5
2000
2001
20.5
20.4
Use the vertical line test. Notice that no vertical
line can be drawn that contains more than one
of the data points.
Answer: Yes, this relation is a function.
Health The table shows
the average weight of a
baby for several months
during the first year.
Graph this information
and determine whether it
represents a function.
Age
(months)
Weight
(pounds)
1
12.5
2
4
6
9
16
22
24
25
12
26
Answer:
Yes, this relation is a function.
Graph the relation represented by
Make a table of values to find ordered pairs that satisfy the
equation. Choose values for x and find the corresponding
values for y. Then graph the ordered pairs.
x
y
–1
0
1
2
–4
–1
2
5
(2, 5)
(1, 2)
(0, –1)
(–1, –4)
Find the domain and range.
Since x can be any real number,
there is an infinite number of
ordered pairs that can be
graphed. All of them lie on the
line shown. Notice that every
real number is the x-coordinate
of some point on the line. Also,
every real number is the
y-coordinate of some point
on the line.
(2, 5)
(1, 2)
(0, –1)
(–1, –4)
Answer: The domain and range are both all
real numbers.
Determine whether the relation is a function.
This graph passes the vertical
line test. For each x value, there
is exactly one y value.
(2, 5)
(1, 2)
Answer: Yes, the equation
represents
a function.
(0, –1)
(–1, –4)
a. Graph
Answer:
b. Find the domain and range.
Answer: The domain and range
are both all real numbers.
c. Determine whether the
relation is a function.
Answer: Yes, the equation
is a function.
Graph the relation represented by
Make a table. In this case, it is easier to choose y values
and then find the corresponding values for x. Then sketch
the graph, connecting the points with a smooth curve.
x
y
5
2
1
2
5
–2
–1
0
1
2
(5, 2)
(2, 1)
(1, 0)
(2, –1)
(5, –2)
Find the domain and range.
Every real number is the
y-coordinate of some point on
the graph, so the range is all real
numbers. But, only real numbers
that are greater than or equal to
1 are x-coordinates of points on
the graph.
Answer: The domain is
The range is all
real numbers.
(5, 2)
(2, 1)
(1, 0)
(2, –1)
(5, –2)
.
Determine whether the relation is a function.
x
y
5
2
1
2
5
–2
–1
0
1
2
(5, 2)
(2, 1)
(1, 0)
(2, –1)
(5, –2)
You can see from the table and the vertical line test that
there are two y values for each x value except x = 1.
Answer: The equation
a function.
does not represent
a. Graph
Answer:
b. Find the domain and range.
Answer: The domain is
{x|x  –3}. The range
is all real numbers.
c. Determine whether the
relation is a function.
Answer: No, the equation
represent a function.
does not
Given
, find
Original function
Substitute.
Simplify.
Answer:
Given
find
Original
function
Substitute.
Multiply.
Simplify.
Answer:
Given
, find
Original function
Substitute.
Answer:
Given
find each value.
and
a.
Answer: 6
b.
Answer: 0.625
c.
Answer:
Example 1 Identify Linear Functions
Example 2 Evaluate a Linear Function
Example 3 Standard Form
Example 4 Use Intercepts to Graph a Line
State whether
Explain.
is a linear function.
Answer: This is a linear function because it is in the form
State whether
Explain.
is a linear function.
Answer: This is not a linear function because x has an
exponent other than 1.
State whether
Explain.
is a linear function.
Answer: This is a linear function because it can be written
as
State whether each function is a linear function.
Explain.
a.
Answer: yes;
b.
Answer: No; x has an exponent
other than 1.
c.
Answer: No; two variables are
multiplied together.
Meteorology The linear function
can be used to find the number of degrees Fahrenheit,
f (C), that are equivalent to a given number of degrees
Celsius, C.
On the Celsius scale, normal body temperature is
37C. What is normal body temperature in degrees
Fahrenheit?
Original function
Substitute.
Simplify.
Answer: Normal body temperature, in degrees
Fahrenheit, is 98.6F.
There are 100 Celsius degrees between the freezing
and boiling points of water and 180 Fahrenheit
degrees between these two points. How many
Fahrenheit degrees equal 1 Celsius degree?
Divide 180 Fahrenheit degrees by 100 Celsius degrees.
Answer: 1.8F = 1C
Meteorology The linear function
can be
used to find the distance d(s) in miles from a storm,
based on the number of seconds s that it takes to
hear thunder after seeing lightning.
a. If you hear thunder 10 seconds after seeing lightning,
how far away is the storm?
Answer: 2 miles
b. If the storm is 3 miles away, how long will it take to
hear thunder after seeing lightning?
Answer: 15 seconds
Write
in standard form. Identify A, B, and C.
Original equation
Subtract 3x from each side.
Multiply each side by –1 so
that A  0.
Answer:
and
Write
and C.
in standard form. Identify A, B,
Original equation
Subtract 2y from each side.
Multiply each side by –3 so that
the coefficients are all integers.
Answer:
and
Write
and C.
in standard form. Identify A, B,
Original equation
Subtract 4 from each side.
Divide each side by 2 so that the
coefficients have a GCF of 1.
Answer:
and
Write each equation in standard form. Identify A, B,
and C.
a.
Answer:
and
b.
Answer:
and
c.
Answer:
and
Find the x-intercept and the y-intercept of the graph of
Then graph the equation.
The x-intercept is the value of x when
Original equation
Substitute 0 for y.
Add 4 to each side.
Divide each side by –2.
The x-intercept is –2. The graph crosses the
x-axis at (–2, 0).
Likewise, the y-intercept is the value of y when
Original equation
Substitute 0 for x.
Add 4 to each side.
The y-intercept is 4. The graph crosses the y-axis at (0, 4).
Use the ordered pairs to graph this equation.
Answer: The x-intercept is –2, and the y-intercept is 4.
(0, 4)
(–2, 0)
Find the x-intercept and the y-intercept of the graph of
Then graph the equation.
Answer: The x-intercept is –2, and the y-intercept is 6.
Example 1 Find Slope
Example 2 Use Slope to Graph a Line
Example 3 Rate of Change
Example 4 Parallel Lines
Example 5 Perpendicular Line
Find the slope of the line that passes through (1, 3)
and (–2, –3). Then graph the line.
Slope formula
and
Simplify.
Graph the two ordered pairs and draw the line.
Use the slope to check your
graph by selecting any point
on the line. Then go up 2 units
and right 1 unit or go down 2
units and left 1 unit. This point
should also be on the line.
Answer: The slope of the
line is 2.
(1, 3)
(–2, –3)
Find the slope of the line that passes through (2, 3)
and (–1, 5). Then graph the line.
Answer: The slope of the
line is
Graph the line passing through (1, –3) with a slope
of
Graph the ordered pair (1, –3).
Then, according to the slope,
go down 3 units and right 4
units. Plot the new point at
(5, –6).
Draw the line containing
the points.
(1, –3)
(5, –6)
Graph the line passing through (2, 5) with a slope
of –3.
Answer:
Communication Refer to
the graph. Find the rate of
change of the number of
radio stations on the air in
the United States from 1990
to 1998.
Slope
formula
Substitute.
Simplify.
Answer: Between 1990 and 1998, the number of radio
stations on the air in the United States increased at an
average rate of 0.225(1000) or 225 stations per year.
Computers Refer to the
graph. Find the rate of
change of the number of
households with computers
in the United States from
1984 to 1998.
Answer: The rate of change
is 2.9 million households
per year.
Graph the line through (1, –2) that is parallel to the
line with the equation
The x-intercept is –2 and the
y-intercept is 2.
Use the intercepts to graph
The line rises 1 unit for every
1 unit it moves to the right, so
the slope is 1.
Now, use the slope and the
point at (1, –2) to graph the
line parallel to
(2, –1)
(1, –2)
Graph the line through (2, 3) that is parallel to the
line with the equation
Answer:
Graph the line through (2, 1) that is perpendicular
to the line with the equation
The x-intercept is
or 1.5
and the y-intercept is –1.
Use the intercepts to graph
2x – 3y = 3
The line rises 1 unit for every
1.5 units it moves to the right,
so the slope is
or
Graph the line through (2, 1) that is perpendicular
to the line with the equation
The slope of the line
perpendicular is the opposite
reciprocal of
or
Start at (2, 1) and go down
3 units and right 2 units.
Use this point and (2, 1) to
graph the line.
(2, 1)
2x – 3y = 3
(4, –2)
Graph the line through (–3, 1) that is perpendicular
to the line with the equation
Answer:
Example 1 Write an Equation Given Slope and a Point
Example 2 Write an Equation Given Two Points
Example 3 Write an Equation for a Real-World Situation
Example 4 Write an Equation of a Perpendicular Line
Write an equation in slope-intercept form for the line
that has a slope of
and passes through (5, –2).
Slope-intercept form
Simplify.
Add 3 to each side.
Answer: The y-intercept is 1. So, the equation in
slope-intercept form is
Write an equation in slope-intercept form for the line
that has a slope of
Answer:
and passes through (–3, –1).
Multiple-Choice Test Item
What is an equation of the line through (2, –3)
and (–3, 7)?
A
B
C
D
Read the Test Item
You are given the coordinates of two points on the line.
Notice that the answer choices are in slope-intercept form.
Solve the Test Item
First, find the slope of the line.
Slope formula
Simplify.
The slope is –2. That eliminates choices
B and C.
Then use the point-slope formula to find an equation.
Point-slope form
you can use either
point for
.
Distributive Property
Subtract 3 from each side.
Answer: D
Multiple-Choice Test Item
What is an equation of the line through (2, 5)
and (–1, 3)?
A
C
B
D
Answer: C
Sales As a part-time salesperson, Jean Stock is paid
a daily salary plus commission. When her sales are
$100, she makes $58. When her sales are $300, she
makes $78.
Write a linear equation to
model this situation.
Let x be her sales and let y be
the amount of money she makes.
Use the points (100, 58) and
(300, 78) to make a graph to
represent the situation.
Slope formula
Simplify.
Now use the slope and either of the given points with the
point-slope form to write the equation.
Point-slope form
Distributive Property
Add 58 to each side.
Answer: The slope-intercept form of the equation
is
What are Ms. Stock’s daily
salary and commission rate?
The y-intercept of the line is 48.
The y-intercept represents the
money Jean would make if she
had no sales. Thus, $48 is her
daily salary.
The slope of the line is 0.1. Since
the slope is the coefficient of x,
which is her sales, she makes
10% commission.
Answer: Ms. Stock’s daily salary is $48, and she makes
a 10% commission.
How much would Jean make in a day if her sales
were $500?
Find the value of y when
Use the equation you found
in Example 3a.
Replace x with 500.
Simplify.
Answer: She would make $98 if her sales were $500.
Sales The student council is selling coupon books to
raise money for the Humane Society. If the group
sells 200 books, they will receive $150 dollars. If they
sell 500 books, they will make $375.
a. Write a linear equation to model this situation.
Answer:
b. Find the percentage of the proceeds that the student
council receives.
Answer: 75%
c. If they sold 1000 books, how much money would
they receive to donate to the Humane Society?
Answer: $750
Write an equation for the line that passes through
(3, –2) and is perpendicular to the line whose
equation is
The slope of the given line is –5. Since the slopes of
perpendicular lines are opposite reciprocals, the slope
of the perpendicular line is
Use the point-slope form and the ordered pair (3, –2)
to write the equation.
Point-slope form
Distributive Property
Subtract 2 from each side.
Answer: An equation of the line is
Write an equation for the line that passes through
(3, 5) and is perpendicular to the line whose
equation is
Answer:
Example 1 Draw a Scatter Plot
Example 2 Find and Use a Prediction Equation
Education The table below shows the approximate
percent of students who sent applications to two
colleges in various years since 1985. Make a scatter
plot of the data.
Years Since
1985
0
3
6
9
12
15
Percent
20
18
15
15
14
13
Source: U.S. News & World Report
Graph the data as ordered pairs,
with the number of years since 1985
on the horizontal axis and the
percentage on the vertical axis.
Safety The table below shows
the approximate percent of
drivers who wear seat belts in
various years since 1994. Make
a scatter plot of the data.
Years Since
1994
0
Percent
57 58 61 64 69 68 71 73
1
2
3
4
5
6
7
Source: National Highway Traffic Safety Administration
Education The table and scatter plot below show
the approximate percent of students who sent
applications to two colleges in various years
since 1985.
Draw a line of fit for the data. How
well does the line fit the data?
Years Since
1985
0
3
6
9
12
15
Percent
20
18
15
15
14
13
Source: U.S. News & World Report
The points (3, 18) and (15, 13)
appear to represent the data well.
Draw a line through these two points.
Education The table and scatter plot below show
the approximate percent of students who sent
applications to two colleges in various years
since 1985.
Draw a line of fit for the data. How
well does the line fit the data?
Years Since
1985
0
3
6
9
12
15
Percent
20
18
15
15
14
13
Source: U.S. News & World Report
Answer: Except for (6, 15), this
line fits the data fairly well.
Find a prediction equation. What do the slope and
y-intercept indicate?
Find an equation of the line through (3, 18) and (15, 13).
Begin by finding the slope.
Slope formula
Substitute.
Simplify.
Point-slope form
Distributive Property
Add 18 to each side.
Answer: One prediction equation is
The slope indicates that the percent of students sending
applications to two colleges is falling at about 0.4% each
year. The y-intercept indicates that the percent in 1985
should have been about 19%.
Predict the percent in 2010.
The year 2010 is 25 years after 1985, so use the
prediction equation to find the value of y when
Prediction equation
Simplify.
Answer: The model predicts that the percent in 2010
should be about 9%.
How accurate is this prediction?
Answer: The fit is only approximate, so the prediction
may not be very accurate.
Safety The table and scatter plot
show the approximate percent of
drivers who wear seat belts in
various years since 1994.
Years Since
1994
0
Percent
57 58 61 64 69 68 71 73
1
2
3
4
5
6
7
Source: National Highway Traffic Safety Administration
a. Draw a line of fit for the data.
How well does the line fit the data?
Answer: Except for (4, 69), this
line fits the data very well.
b. Find a prediction equation. What do the slope and
y-intercept indicate?
Answer: Using (1, 58) and (7, 73), an equation is
y = 2.5x + 55.5. The slope indicates that the percent of
drivers wearing seatbelts is increasing at a rate of 2.5%
each year. The y-intercept indicates that, according to the
trend of the rest of the data, the percent of drivers who
wore seatbelts in 1994 was about 56%.
c. Predict the percent of drivers who will be wearing
seat belts in 2005.
Answer: 83%
d. How accurate is the prediction?
Answer: Except for the outlier, the line fits the data very
well, so the predicted value should be fairly accurate.
Example 1 Step Function
Example 2 Constant Function
Example 3 Absolute Value Functions
Example 4 Piecewise Function
Example 5 Identify Functions
Psychology One psychologist charges for
counseling sessions at the rate of $85 per hour or
any fraction thereof. Draw a graph that represents
this solution.
Explore
The total charge must be a multiple of $85,
so the graph will be the graph of a step
function.
Plan
If the session is greater than 0 hours, but
less than or equal to 1 hour, the cost is
$85. If the time is greater than 1 hour, but
less than or equal to 2 hours, then the cost
is $170, and so on.
Solve
Use the pattern of times and costs to make a
table, where x is the number of hours of the
session and C(x) is the total cost. Then draw
the graph.
x
C(x)
85
170
255
340
425
Answer:
Examine Since the psychologist rounds any fraction of
an hour up to the next whole number, each
segment on the graph has a circle at the left
endpoint and a dot at the right endpoint.
Sales The Daily Grind charges $1.25 per pound of
meat or any fraction thereof. Draw a graph that
represents this situation.
Answer:
Graph
For every value of
horizontal line.
g(x) = –3
x
g(x)
–2
–3
0
1
0.5
–3
–3
–3
The graph is a
Answer:
Graph
Answer:
Graph
and
on the same
coordinate plane. Determine the similarities and
differences in the two graphs.
Find several ordered pairs for each function.
x
|x–3|
x
|x+2|
0
3
–4
2
1
2
–3
1
2
1
–2
0
3
0
–1
1
4
1
0
2
5
2
1
3
Graph the points and connect them.
Answer:
• The domain of both graphs
is all real numbers.
• The range of both graphs
is
• The graphs have the
same shape, but different
x-intercepts.
• The graph of g(x) is the
graph of f(x) translated
left 5 units.
Graph
and
on the same
coordinate plane. Determine the similarities and
differences in the two graphs.
Answer:
• The domain of both graphs
is all real numbers.
• The range of
is
The range of
is
• The graphs have the same
shape, but different y-intercepts.
• The graph of g(x) is the graph of f(x)
translated up 5 units.
Graph
and range.
Identify the domain
Step 1 Graph the linear function
for
Since 3 satisfies this
inequality, begin with a
closed circle at (3, 2).
Graph
and range.
Identify the domain
Step 2 Graph the constant
function
Since x does
not satisfy this
inequality, begin with an
open circle at (3, –1)
and draw a horizontal
ray to the right.
Graph
and range.
Identify the domain
Answer: The function is defined
for all values of x, so the domain
is all real numbers. The values
that are y-coordinates of points
on the graph are all real numbers
less than or equal to –2, so the
range is
Graph
and range.
Answer: The domain is all real
numbers. The range
is
Identify the domain
Determine whether the graph
represents a step function,
a constant function, an
absolute value function,
or a piecewise function.
Answer: Since this graph
consists of different
rays and segments,
it is a piecewise
function.
Determine whether the graph
represents a step function,
a constant function, an
absolute value function,
or a piecewise function.
Answer: Since this graph is
V-shaped, it is an
absolute value
function.
Determine whether each graph represents a step
function, a constant function, an absolute value
function, or a piecewise function.
a.
b.
Answer: constant
function
Answer: absolute value
function
Example 1 Dashed Boundary
Example 2 Solid Boundary
Example 3 Absolute Value Inequality
Graph
The boundary is the graph
of
Since the
inequality symbol is <, the
boundary will be dashed.
Use the slope-intercept
form,
Graph
Test (0, 0).
Original
inequality
true
Shade the region that contains (0, 0).
Graph
Answer:
Education The SAT has two parts. One tutoring
company advertises that it specializes in helping
students who have a combined score on the SAT
that is 900 or less.
Write an inequality to describe the combined scores
of students who are prospective tutoring clients.
Let x be the first part of the SAT and let y be the
second part. Since the scores must be 900 or
less, use the  symbol.
The
1st part
x
Answer:
and
2nd part
together
are less than
or equal to
900.
y

900
Graph the inequality.
Since the inequality
symbol is , the graph
of the related linear
equation
is solid. This is the
boundary of the inequality.
Graph the inequality.
Test (0, 0).
Original
inequality
true
Graph the inequality.
Shade the region that
contains (0, 0). Since the
variables cannot be
negative, shade only the
part in the first quadrant.
Does a student with a verbal score of 480 and a math
score of 410 fit the tutoring company’s guidelines?
The point (480, 410) is in
the shaded region, so it
satisfies the inequality.
Answer: Yes, this student fits the tutoring
company’s guidelines.
Class Trip Two social studies classes are going
on a field trip. The teachers have asked for parent
volunteers to also go on the trip as chaperones.
However, there is only enough seating for 60 people
on the bus.
a. Write an inequality to describe the number of
students and chaperones that can ride on the bus.
Answer:
b. Graph the inequality.
Answer:
c. Can 45 students and 10 chaperones go on the trip?
Answer: yes
Graph
Since the inequality
symbol is , the graph
of the related equation
is solid.
Graph the equation.
Test (0, 0).
Original inequality
true
Shade the region that
contains (0, 0).
Graph
Answer:
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