Transcript Chapter 8 - Crestwood Local Schools

Relations and Functions
PRE-ALGEBRA LESSON 8-1
Determine if and where Will and Pedro will meet if they start
driving from the same intersection. Pedro travels 2 blocks east, 2
blocks south, 2 blocks east, and 2 blocks north. Will starts traveling
2 blocks north, 4 blocks east, and 2 blocks south.
Yes, they will meet 4 blocks east.
8-1
Relations and Functions
PRE-ALGEBRA LESSON 8-1
(For help, go to Lesson 1-10.)
Graph each point.
1. A(3, 4)
2. B(–3, 1)
3. F(2, 0)
4. D(2, –2)
5. C(–4, –3)
6. E(0, –4)
Check Skills You’ll Need
8-1
Relations and Functions
PRE-ALGEBRA LESSON 8-1
Solutions
8-1
Relations and Functions
PRE-ALGEBRA LESSON 8-1
Is each relation a function? Explain.
a. {(0, 5), (1, 6), (2, 4), (3, 7)}
List the domain values and the range values
in order.
Draw arrows from the domain values to their
range values.
There is one range value for each domain value. This relation is a
function.
8-1
Relations and Functions
PRE-ALGEBRA LESSON 8-1
(continued)
b. {(0, 5), (1, 5), (2, 6), (3, 7)}
There is one range value for each domain value. This relation is a
function.
8-1
Relations and Functions
PRE-ALGEBRA LESSON 8-1
(continued)
c. {(0, 5), (0, 6), (1, 6), (2, 7)}
There are two range values for the domain value 0.
This relation is not a function.
Quick Check
8-1
Relations and Functions
PRE-ALGEBRA LESSON 8-1
Is the time needed to mow a lawn a function of the
size of the lawn? Explain.
No; two lawns of the same size (domain value)
can require different lengths of time (range values)
for mowing.
Quick Check
8-1
Relations and Functions
PRE-ALGEBRA LESSON 8-1
a. Graph the relation shown in the table.
Domain
Value
–3
–5
3
5
Graph the ordered pairs
(–3, 5), (–5, 3), (3, 5), and
(5, 3).
Range
Value
5
3
5
3
8-1
Relations and Functions
PRE-ALGEBRA LESSON 8-1
(continued)
b. Use the vertical-line test. Is the relation a function? Explain.
Pass a pencil across the graph as shown.
Keep the pencil vertical (parallel to the
y-axis) to represent a vertical line.
The pencil does not pass through two points at any one of its
positions, so the relation is a function.
Quick Check
8-1
Relations and Functions
PRE-ALGEBRA LESSON 8-1
Is each relation a function? Explain.
1. {(–2, –1), (4, 2), (–8, –4), (6, 3)}
Yes; there is only one range value for each domain value.
2. {(5, 0), (7, 2), (9, 4), (5, 1)}
No; the domain value 5 has two range values, 0 and 1.
3. Graph the relation in the table.
Is the relation a function?
Explain.
Yes; there is one range value for
each domain value. Check students’ graphs.
8-1
x
–1
y
7
0
1
2
7
7
7
Equations With Two Variables
PRE-ALGEBRA LESSON 8-2
Identify the data needed: At a fundraiser, $5 was collected from
each parent and $1 was collected from each child. How much
money was raised?
the number of parents and children at the fundraiser
8-2
Equations With Two Variables
PRE-ALGEBRA LESSON 8-2
(For help, go to Lesson 1-3.)
Evaluate each expression for x = 2.
1. 2 + x
2. x – 12
3. 8x – 13
4. 24 ÷ 2x
Check Skills You’ll Need
8-2
Equations With Two Variables
PRE-ALGEBRA LESSON 8-2
Solutions
1. 2 + x = 2 + 2
=4
2. x – 12 = 2 – 12
= –10
3. 8x – 13 = 8(2) – 13
= 16 – 13
=3
4. 24 ÷ 2x = 24 ÷ 2(2)
= 24 ÷ 4
=6
8-2
Equations With Two Variables
PRE-ALGEBRA LESSON 8-2
Find the solution of y = 4x – 3 for x = 2.
y = 4x – 3
y = 4(2) – 3
y=8–3
y=5
Replace x with 2.
Multiply.
Subtract.
A solution of the equation is (2, 5).
Quick Check
8-2
Equations With Two Variables
PRE-ALGEBRA LESSON 8-2
The equation a = 5 + 3p gives the price for
admission to a park. In the equation, a is the admission
price for one car with p people in it. Find the price of
admission for a car with 4 people in it.
a = 5 + 3p
a = 5 + 3(4)
a = 5 + 12
a = 17
Replace p with 4.
Multiply.
Add.
A solution of the equation is (4, 17). The admission price for one
car with 4 people in it is $17.
Quick Check
8-2
Equations With Two Variables
PRE-ALGEBRA LESSON 8-2
Graph y = 4x – 2.
Make a table of values to show ordered-pair solutions.
x
–2
0
2
4x – 2
4(–2) – 2 = –8 – 2 = –10
4(0) – 2 = 0 – 2 = –2
(x, y)
(–2, –10)
(0, –2)
4(2) – 2 = 8 – 2 = 6
(2, 6)
Graph the ordered pairs.
Draw a line through the points.
Quick Check
8-2
Equations With Two Variables
PRE-ALGEBRA LESSON 8-2
Graph each equation. Is the equation a function?
a. y = –3
b. x = 4
For every value of x, y = –3.
For every value of y, x = 4.
This is a horizontal line.
The equation y = –3 is a
function.
This is a vertical line.
The equation y = 4
is not a function.
8-2
Quick Check
Equations With Two Variables
PRE-ALGEBRA LESSON 8-2
Solve y – 1 x = 3 for y. Then graph the equation.
2
Solve the equation for y.
y–
1
1
x=3
2
1
1
y – 2x + 2x = 3 + 2 x
1
y=2x+3
1
Add 2 x to each side.
Simplify.
8-2
Equations With Two Variables
PRE-ALGEBRA LESSON 8-2
(continued)
Make a table of values.
1
2
1
–2
2
1
0
2
1
2
2
x
Graph.
x+3
(x, y)
(–2) + 3 = –1 + 3 = 2
(–2, 2)
(0) + 3 = 0 + 3 = 3
(0, 3)
(2) + 3 = 1 + 3 = 4
(2, 4)
Quick Check
8-2
Equations With Two Variables
PRE-ALGEBRA LESSON 8-2
Find the solution for each equation for x = 2.
1. y = –2x + 5
2. y = 7x
(2, 1)
(2, 14)
3. y = 3x – 9
(2, –3)
Solve each equation for y. Then graph each equation.
4. y – 2x = 3
5. 2x + 2y = 8
y = 2x + 3
y = –x + 4
8-2
Slope and y-intercept
PRE-ALGEBRA LESSON 8-3
Floyd has $35 in the bank. He writes a check for $52 and makes a
deposit of $10. What is his new balance?
– $7
8-3
Slope and y-intercept
PRE-ALGEBRA LESSON 8-3
(For help, go to Lesson 1-6.)
Find each difference.
1. –4 – 5
2. 3 – (–2)
3. 6 – 9
4. –1 – (–1)
Check Skills You’ll Need
8-3
Slope and y-intercept
PRE-ALGEBRA LESSON 8-3
Solutions
1. –4 – 5 = –4 + (–5)
= –9
2. 3 – (–2) = 3 + 2
=5
3. 6 – 9 = 6 + (–9)
=–3
4. –1 – (–1) = –1 + 1
=0
8-3
Slope and y-intercept
PRE-ALGEBRA LESSON 8-3
Find the slope of each line.
a.
b.
rise
4
rise
–6
slope = run = 3 = –2
slope = run = 1 = 4
Quick Check
8-3
Slope and y-intercept
PRE-ALGEBRA LESSON 8-3
Find the slope of the line through E(7, 5) and F(–2, 0).
difference in y-coordinates
0–5
–5
5
slope = difference in x-coordinates = –2 – 7 = –9 = 9
Quick Check
8-3
Slope and y-intercept
PRE-ALGEBRA LESSON 8-3
Find the slope of each line.
a.
b.
–3 – (–3)
–1 – 3
0
–4
slope = 4 – (–2) = 6 = 0
slope = –2 – (–2) = 0
Slope is 0 for a horizontal line.
Division by zero is
undefined. Slope is
undefined for a vertical line.
Quick Check
8-3
Slope and y-intercept
PRE-ALGEBRA LESSON 8-3
A ramp slopes from a warehouse door down to a
street. The function y = – 1 x + 4 models the ramp, where x is
5
the distance in feet from the bottom of the door and y is the
height in feet above the street. Graph the equation.
Step 1
Since the y-intercept is 4,
graph (0, 4).
Step 2
Since the slope is – 5 , move
1 unit down from (0, 4).
Then move 5 units right to
graph a second point.
Step 3
Draw a line through the
points.
1
Quick Check
8-3
Slope and y-intercept
PRE-ALGEBRA LESSON 8-3
Find the slope of the line through each pair of points.
1. A(2, 4), B(–2, –4)
2
2. F(–5, 1), G(0, –9)
–2
3. Identify the slope and y-intercept of y = – 4 x + 3.
3
Then graph the line.
–
4
;3
3
8-3
Writing Rules for Linear Functions
PRE-ALGEBRA LESSON 8-4
Find the slope of each line.
a.
10x + 5y = 28
–2
b.
3x +3y = 5
–1
8-4
Writing Rules for Linear Functions
PRE-ALGEBRA LESSON 8-4
(For help, go to Lesson 8-3.)
Find the slope of the line through each pair of points.
1. A(3, 1), B(2, 1)
2. S(3, 4), T(1, 2)
3. P(0, –2), Q(0, 2)
4. C(–5, 2), D(4, –1)
Check Skills You’ll Need
8-4
Writing Rules for Linear Functions
PRE-ALGEBRA LESSON 8-4
Solutions
0
1–1
= –1 = 0
2–3
–2
2–4
2. slope = 1 – 3 = –2 = 1
4
2 – (–2)
3. slope = 0 – 0 = 0 ; slope is undefined.
–3
–1 – 2
1
4. slope = 4 – (–5) = 9 = – 3
1. slope =
8-4
Writing Rules for Linear Functions
PRE-ALGEBRA LESSON 8-4
A long-distance phone company charges its
customers a monthly fee of $4.95 plus 9¢ for each minute of
a long-distance call.
a.
Write a function rule that relates the total monthly bill to the
number of minutes a customer spent on long-distance calls.
Words
total bill
is
$4.95
plus
9¢
times
number of
minutes
Let m = the number of minutes.
Let t( m ) = total bill, a function of the number of minutes.
Rule
t( m )
=
4.95
+
0.09
A rule for the function is t(m) = 4.95 + 0.09m.
8-4
•
m
Writing Rules for Linear Functions
PRE-ALGEBRA LESSON 8-4
(continued)
b.
Find the total monthly bill if the customer made 90 minutes
of long-distance calls.
t(m) = 4.95 + 0.09m
t(90) = 4.95 + 0.09(90)
Replace m with 90.
t(90) = 4.95 + 8.10
Multiply.
t(90) = 13.05
Add.
The total monthly bill with 90 minutes of long-distance calls
is $13.05.
Quick Check
8-4
Writing Rules for Linear Functions
PRE-ALGEBRA LESSON 8-4
Write a rule for the linear function in the
table below.
–2
–2
–2
x
f(x)
2
0
–2
–4
3
–5
–13
–21
As the x values decrease by 2,
the f(x) values decrease by 8.
–8
–8
–8
–8
So m = –2 = 4.
When x = 0, f(x) = –5. So b = –5.
A rule for the function is f(x) = 4x – 5.
Quick Check
8-4
Writing Rules for Linear Functions
PRE-ALGEBRA LESSON 8-4
Write a rule for the linear function graphed below.
–2 – 2
–4
slope = 0 – 2 = –2 = 2
y-intercept = –2
A rule for the function is f(x) = 2x – 2.
Quick Check
8-4
Writing Rules for Linear Functions
PRE-ALGEBRA LESSON 8-4
Write a rule for each function.
1. Sarena earns a salary of $150 a week plus a 10%
commission on each sale.
f(x) = 0.10x + 150
2.
x
0
f(x)
0
1
2
1
2
3.
f(x) = x
f(x) = 5x – 3
8-4
Scatter Plots
PRE-ALGEBRA LESSON 8-5
Use a graph with 100 squares to draw and shade squares that spell a
3-letter word. Estimate what fraction of the grid is unshaded.
Check students’ answers.
8-5
Scatter Plots
PRE-ALGEBRA LESSON 8-5
(For help, go to Lesson 1-10.)
Write the coordinates of each point.
1. A
2. B
3. C
4. D
Check Skills You’ll Need
8-5
Scatter Plots
PRE-ALGEBRA LESSON 8-5
Solutions
1. A(–2, 2)
2. B(0, 3)
3. C(–3, 0)
4. D(2, 3)
8-5
Scatter Plots
PRE-ALGEBRA LESSON 8-5
The scatter plot shows education and income data.
a.
Describe the person represented by the point
with coordinates (10, 30).
This person has 10 years of education and
earns $30,000 each year.
b.
How many people have exactly 14 years of
education? What are their incomes?
The points (14, 50), (14, 80), and (14, 90)
have education coordinate 14.
The three people they represent earn
$50,000, $80,000, and $90,000, respectively.
Quick Check
8-5
Scatter Plots
PRE-ALGEBRA LESSON 8-5
Use the table to make a scatter plot of the
elevation and precipitation data.
Elevation Above
Sea Level (ft)
1,050
Atlanta, GA
20
Boston, MA
596
Chicago, IL
18
Honolulu, HI
11
Miami, FL
1,072
Phoenix, AZ
75
Portland, ME
40
San Diego, CA
1,305
Wichita, KS
City
Mean Annual
Precipitation (in.)
51
42
36
22
56
8
44
10
29
8-5
Quick Check
Scatter Plots
PRE-ALGEBRA LESSON 8-5
Use the scatter plot below. Is there a positive
correlation, a negative correlation, or no correlation
between temperatures and amounts of precipitation?
Explain.
The values show no relationship.
There is no correlation.
Quick Check
8-5
Scatter Plots
PRE-ALGEBRA LESSON 8-5
Answer the following questions based on the graph.
1. What do you know about the student at
point A?
read 2 books per month, grade: 80
2. How many students read 3 books
per month?
3
3. Is there a positive correlation, a negative correlation, or no
correlation between books read and semester grades?
positive correlation
8-5
Problem Solving Strategy: Solve by Graphing
PRE-ALGEBRA LESSON 8-6
Draw 6 nonlinear points. How many different segments can
you draw connecting two points?
15
8-6
Problem Solving Strategy: Solve by Graphing
PRE-ALGEBRA LESSON 8-6
(For help, go to Lesson 8-4.)
Write a rule for each linear function.
1.
2.
Check Skills You’ll Need
8-6
Problem Solving Strategy: Solve by Graphing
PRE-ALGEBRA LESSON 8-6
Solutions
1. slope = 6 – 2 = 4 = 2
2–0
2
y-intercept = 2
A rule for the function is f(x) = 2x + 2.
2. slope = –2 – (– 3) = 1 = – 1
–4 – 0
–4
4
y-intercept = –3
A rule for the function is f(x) = – 1 x – 3.
4
8-6
Problem Solving Strategy: Solve by Graphing
PRE-ALGEBRA LESSON 8-6
Use the data in the table below. Suppose this year
there are 16 wolves on the island. Predict how many moose
are on the island.
Isle Royale Populations
Year
Wolf
1982
1983
1984
14
23
24
1985
1986
1987
22
20
16
Moose
Year
Wolf
Moose
Year
Wolf
Moose
700
900
811
1988
1989
1990
12
11
15
1,653
1,397
1,216
1994
1995
1996
15
16
22
1,800
2,400
1,200
1,062
1,025
1,380
1991
1992
1993
12
12
13
1,313
1,600
1,880
1997
1998
1999
24
14
25
500
700
750
8-6
Problem Solving Strategy: Solve by Graphing
PRE-ALGEBRA LESSON 8-6
(continued)
Step 1
Make a scatter plot by
graphing the (wolf, moose)
ordered pairs. Use the x-axis
for wolves and the y-axis for
moose.
Step 2
Sketch a trend line. The line
should be as close as
possible to each data point.
There should be about as
many points above the trend
line as below it.
8-6
Problem Solving Strategy: Solve by Graphing
PRE-ALGEBRA LESSON 8-6
(continued)
Step 3
To predict the number of
moose when there are 16
wolves, find 16 along the
horizontal axis.
Look up to find the point on
the trend line that
corresponds to 16 wolves.
Then look across to the value
on the vertical axis, which is
about 1,300.
There are about 1,300 moose on the
island.
Quick Check
8-6
Problem Solving Strategy: Solve by Graphing
PRE-ALGEBRA LESSON 8-6
Answer each question.
1. When is there no trend line for a scatter plot?
when there is no correlation
2. What type of correlation will the data have for a scatter plot of
ages and heights of all students in your school?
positive
3. Why is it important to say “about” when making a prediction?
The prediction is based on a trend line that approximates the
locations of the related points.
8-6
Solving Systems of Linear Equations
PRE-ALGEBRA LESSON 8-7
Find the slope and y-intercept for the equation 1.3x – y + 5 = 0.
Slope is 1.3; y-intercept is 5.
8-7
Solving Systems of Linear Equations
PRE-ALGEBRA LESSON 8-7
(For help, go to Lesson 8-2.)
Graph each equation.
1. y = –x – 4
2. y = 2x – 1
3. –4x = 6y
4. 3x – 2y = 5
Check Skills You’ll Need
8-7
Solving Systems of Linear Equations
PRE-ALGEBRA LESSON 8-7
Solutions
1. y = –x + 4
2. y = 2x – 1
8-7
Solving Systems of Linear Equations
PRE-ALGEBRA LESSON 8-7
Solutions (continued)
3.
–4x = 6y
– 4x=y
6
y=–2x
4.
3
3x – 2y = 5
3x – 2y – 3x = 5 – 3x
–2y = –3x + 5
y = 3x – 5
2
8-7
2
Solving Systems of Linear Equations
PRE-ALGEBRA LESSON 8-7
Solve the system y = x – 7 and y = 4x + 2 by
graphing.
Step 1 Graph each line.
Step 2 Find the point of intersection.
The lines intersect at one point, (–3, –10).
The solution is (–3, –10).
Check
See whether (–3, –10) makes both equations
true.
y=x–7
–10
–3 – 7
–10 = –10
Replace x with – 3
and y with –10.
The solution checks.
y = 4x + 2
–10
4(–3) + 2
–10 = –10
Quick Check
8-7
Solving Systems of Linear Equations
PRE-ALGEBRA LESSON 8-7
Quick Check
Solve each system of equations by graphing.
a. 27x + 9y = 36; y = 4 – 3x
b. 8 = 4x + 2y; 2x + y = 5
The lines are the same line.
There are infinitely
many solutions.
The lines are parallel.
They do not intersect.
There is no solution.
8-7
Solving Systems of Linear Equations
PRE-ALGEBRA LESSON 8-7
Find two numbers with a sum of 10 and a
difference of 2.
Step 1 Write equations.
Let x = the greater number.
Let y = the lesser number.
Equation 1 Sum
x+y
is
=
Equation 2 Difference
x–y
10.
10
is
=
Step 2 Graph the equations.
The lines intersect at (6, 4).
The numbers are 6 and 4.
8-7
2.
2
Solving Systems of Linear Equations
PRE-ALGEBRA LESSON 8-7
(continued)
Check Since the sum of 6 and 4 is 10 and the difference of 6 and 4 is 2,
the answer is correct.
Quick Check
8-7
Solving Systems of Linear Equations
PRE-ALGEBRA LESSON 8-7
Solve each system by graphing.
1. y = –x – 4 and y = 4x + 1
(–1, –3)
2. y = –3x + 12 and 18x + 6y = 42
no solution
3. Find two numbers with a sum of 15 and a
difference of 1. Show your work.
(8, 7)
8-7
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
Use two of the given digits to make the equation true. 2, 3, 4
8÷
?
= 10 2
?
3
3
4
8-8
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
(For help, go to Lesson 2-9.)
Is the given value of x a solution of the inequality? Explain.
1. x + 3 <
– –2; x = –5
2. 5 – x < 4; x = 1
3. –2 – 2x <
– 6; x = –4
4. 4x + 1 > –7; x = –2
Check Skills You’ll Need
8-8
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
Solutions
1. yes; x + 3 >
– –2; x = –5
–5 + 3 >
– –2
–2 >
– –2 true
2. no; 5 – x < 4; x = 1
5–1<4
4 < 4 false
3. yes; –2 – 2x <
– 6; x = –4
–2 – 2(–4) <
– 6
–2 + 8 <
–6
6<
– 6 true
4. no; 4x + 1 > –7; x = –2
4(–2) + 1 > – 7
–8 + 1 > –7
–7 > –7 false
8-8
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
Graph each inequality on a coordinate plane.
a.
y > 2x + 1
Step 1 Graph the boundary line.
Points on the boundary line do not
make y > 2x + 1 true. Use a dashed
line.
8-8
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
(continued)
Step 2 Test a point not on the boundary line.
Test (0, 0) in the inequality.
y > 2x + 1
?
0 >? 2(0) + 1
Substitute.
0> 0+1
0>1
false
Since the inequality is false for (0, 0), shade
the region that does not contain (0, 0).
8-8
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
(continued)
b.
y<
– 3x – 2
Step 1 Graph the boundary line.
Points on the boundary line make
y  3x – 2 true. Use a solid line.
8-8
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
(continued)
Step 2 Test a point not on the boundary line.
Test (3, 0) in the inequality.
y<
–? 3x – 2
0<
Substitute.
–? 3(3) – 2
0<
–9–2
0<
true
–7
Since the inequality is true for (3, 0), shade
the region containing (3, 0).
Quick Check
8-8
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
Cashews cost $2/lb. Pecans cost $4/lb. You plan
to spend no more than $20. How many pounds of each can
you buy?
Step 1 Write an inequality.
Words
Inequality
cost of
cashews
plus
cost of
pecans
is at
most
Let
y = number of pounds of cashews.
Let
x = number of pounds of pecans.
2y
+
8-8
4x
<
–
twenty
dollars
20
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
(continued)
Step 2 Write the equation of the boundary line in slopeintercept form.
2y + 4x <
– 20
y<
– –2x + 10
y = –2x + 10
Step 3
Graph y = –2x + 10 in
Quadrant I since weight is not
negative.
8-8
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
(continued)
Step 4 Test (1, 1).
y<
–? –2x + 10
1<
– –2(1) + 10
1<
–8
The inequality is true. (1, 1) is a solution.
Step 5 Shade the region containing (1, 1).
The graph shows the possible solutions. For
example, you could buy 1 pound of pecans and 5
pounds of cashews.
Quick Check
8-8
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
Solve the system y >
– x + 1 and y < 2x + 3 by graphing.
Step 1
Graph y >
– x + 1 on a coordinate
plane. Shade in red.
Step 2
Graph y < 2x + 3 on the same
coordinate plane. Shade in blue.
8-8
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
(continued)
The solutions are the coordinates of all the points in the region that is
shaded in both colors.
Check
See whether the solution (2, 5) makes both of the inequalities
true.
y>
–? x + 1
5>
Replace x with 2 and y with 5.
–2+1
5>
The solution checks.
– 3y
y < 2x + 3
?
5 < 2(2) + 3
5 < 7y
Replace x with 2 and y with 5.
The solution checks.
Quick Check
8-8
Graphing Linear Inequalities
PRE-ALGEBRA LESSON 8-8
Tell whether the ordered pair is a solution of the inequality.
Justify your response.
2. y > –x – 1; (2, –3)
1. y < 4x; (3, 0)
No; –3 > –3.
Yes; 0 < 12.
3. y <
– 4x – 2; (–1, 1)
No; 1 <
– 6.
4. Solve the system y <
– 3x + 1 and
y > –2x + 2 by graphing.
8-8