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Pre-Algebra Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 8-1
Functions
Lesson 8-2
Linear Equations in Two Variables
Lesson 8-3
Graphing Linear Equations Using Intercepts
Example 1 Ordered Pairs and Tables as Functions
Example 2 Use a Graph to Identify Functions
Example 3 Use a Function to Describe Data
Determine whether the relation is a function. Explain.
{(–3, –3), (–1, –1), (0, 0), (–1, 1), (3, 3)}
Answer: No; –1 in the domain is paired with both –1
and 1 in the range.
Determine whether the relation is a function. Explain.
x
7
6
5
2 –3 –6
y
2
4
6
4
2 –2
Answer: Yes; each x value is paired with only one
y value.
Determine whether each relation is a function. Explain.
a. {(2, 5), (4, –1), (3, 1), (6, 0), (–2, –2)}
Answer: Yes; each x value is paired with only
one y value.
b.
x 3
1 –1 –3
y 5
4
3
–4
1 –5
2
1
Answer: No; 1 in the domain is paired with 4 and 2 in
the range.
Determine whether the graph is a function. Explain.
Answer: Yes; it passes the vertical line test.
Determine whether the graph is a function. Explain.
Answer: No; it does not pass the vertical line test.
Business The table shows the number of boxes made.
Number of Hours
Number of Boxes
0
10
20
30
0
3000
6000
9000
Do these data represent a function? Explain.
Answer: Yes; for each 10 hours, only one amount of
boxes is made.
Describe how box production is related to hours
of operation.
Number of Hours Number of Boxes
0
10
20
30
0
3000
6000
9000
Answer: As the number of hours increases, the number
of boxes produced increases.
Assembly Line The table shows the number of
chairs made.
Number of Hours
Number of Chairs
5
10
15
20
120
240
360
480
a. Do these data represent a function? Explain.
Answer: Yes; for each 5 hours, only one amount of
chairs is made.
Assembly Line The table shows the number of
chairs made.
Number of Hours
Number of Chairs
5
10
15
20
120
240
360
480
b. Describe how chair production is related to
hours of operation.
Answer: As hours increase, the number of chairs
produced increases.
Example 1 Find Solutions
Example 2 Solve an Equation for y
Example 3 Graph a Linear Equation
Find four solutions of
.
Choose four values for x. Then substitute each value into
the equation to solve for y. There are many possible
solutions. The solutions you find depend on which x values
you choose.
x
0
1
2
3
y
3
7
11
15
(x, y)
(0, 3)
(1, 7)
(2, 11)
(3, 15)
Sample Answer: Four possible solutions are (0, 3), (1, 7),
(2, 11), and (3, 15).
Find four solutions of
.
Sample Answer: (0, –4), (1, –2), (2, 0), and (3, 2).
Business At a local software company, Level 1
employees x earn $48,000 and Level 2
employees y earn $24,000. Find four solutions of
to determine how
many employees at each level the company can hire
for $216,000.
First, rewrite the equation by solving for y.
Write the equation.
Subtract 48,000x
from each side.
Divide each side
by 24,000.
Simplify.
Choose four x values and substitute them into
x
0
1
2
3
y
9
7
5
3
(x, y)
(0, 9)
(1, 7)
(2, 5)
(3, 3)
Sample Answer: (0, 9), (1, 7), (2, 5), and (3, 3)
0 Level 1, 9 Level 2
1 Level 1, 7 Level 2
2 Level 1, 5 Level 2
3 Level 1, 3 Level 2
Bookstore At a local bookstore, hardbacks are on
sale for $6 and paperbacks are on sale for $3. Bob
has $42 to spend on books. Find four solutions to
determine how many books of each type Bob can
buy with his $42.
Sample Answer: (0, 14), (1, 12), (2, 10), and (3, 8)
0 hardbacks, 14 paperbacks
1 hardbacks, 12 paperbacks
2 hardbacks, 10 paperbacks
3 hardbacks, 8 paperbacks
Graph
by plotting ordered pairs.
First, find ordered pair solutions.
x
–1
0
1
2
y
–4
–3
–2
–1
(x, y)
(–1, –4)
(0, –3)
(1, –2)
(2, –1)
Four solutions are (–1,–4), (0,–3), (1,–2), and (2,–1).
Plot these ordered pairs and draw a line through them.
Note that the ordered pair for any point on this line is a
solution of
. The line is a complete graph of
the function.
Answer:
Check It appears from the graph that (4, 1) is also a
solution. Check this by substitution.
Write the equation.
Replace x with 4 and y with 1.
Simplify.
Graph
Answer:
by plotting ordered pairs.
Example 1 Find Intercepts from Graphs
Example 2 Find Intercepts from Equations
Example 3 Use Intercepts to Graph Equations
Example 4 Intercepts of Real-World Data
Example 5 Horizontal and Vertical Lines
State the x-intercept and the
y-intercept of the line.
The graph crosses the x-axis at
(–2, 0). The x-intercept is –2.
The graph crosses the y-axis at
(0, –2). The y-intercept is –2.
Answer: –2; –2
State the x-intercept and the
y-intercept of the line.
The graph does not cross the
x-axis. There is no x-intercept.
The graph crosses the y-axis
at (0, 2). The y-intercept is 2.
Answer: none; 2
State the x-intercept and the y-intercept of each line.
a.
b.
Answer: 1; 3
Answer: –1; none
Find the x-intercept and the y-intercept for the
graph of
.
To find the x-intercept, let
.
Write the equation.
Replace y with 0.
Simplify.
The x-intercept is 4. So, the graph crosses the
x-axis at (4, 0).
To find the y-intercept, let
.
Write the equation.
Replace x with 0.
Simplify.
Divide each side by 2.
The y-intercept is 2. So, the graph crosses the y-axis
at (0, 2).
Answer: x-intercept, 4; y-intercept, 2
Find the x-intercept and the y-intercept for the graph of
.
Answer: x-intercept, 9; y-intercept, 3
Graph
using the x- and y-intercepts.
Step 1 Find the x-intercept.
Write the equation.
Replace y with 0.
Add 6 to each side.
Divide each side by 3.
The x-intercept is 2, so the graph passes through (2, 0).
Step 2 Find the y-intercept.
Write the equation.
Replace x with 0.
Simplify.
The y-intercept is –6, so the graph passes through (0, –6).
Step 3 Graph the points (2, 0) and (0, –6) and draw a line
through them.
Check Choose some other point on the line and determine
whether its ordered pair is a solution of
.
Answer:
Graph
Answer:
using the x- and y-intercepts.
Home Repair Renata Jones has $300 for home repairs.
A plumber charges her $50 an hour. The equation
represents the amount of money left in
her budget after x number of plumbing hours.
Use the intercepts to graph the equation.
Step 1 Find the x-intercept.
Write the equation.
Replace y with 0.
Subtract 300 from each side.
Simplify.
Divide each side by –50.
The x-intercept is 6.
Step 2 Find the y-intercept.
Write the equation.
Replace x with 0.
Simplify.
The y-intercept is 300.
Step 3 Plot the points with coordinates (6, 0) and (0, 300).
Then draw a line through the points.
Answer:
(0, 300)
(6, 0)
Describe what the intercepts mean.
Answer: The y-intercept shows how much money
Renata has in her budget before any work is done.
The x-intercept means that she can afford only 6
hours of plumbing repair.
Answer:
Shopping Suzanne has
$125 to spend on new
shoes. The shoe store is
having a sale in which
every pair of shoes is
priced at $25. The
equation
represents the amount of
money Suzanne has left
after purchasing x pairs of
shoes.
a. Use the intercepts to
graph the equation.
b. Describe what the intercepts mean.
Answer: The y-intercept shows how much money Suzanne
has before buying any shoes. The x-intercept means that
she can afford only 5 pairs of shoes.
Graph
Note that
using the x- and y-intercepts.
is the same as
The y-intercept is –4, and there is no x-intercept.
Answer:
Graph
Note that
using the x- and y-intercepts.
is the same as
The x-intercept is 5, and there is no y-intercept.
Answer:
Graph each equation using the x- and y-intercepts.
a.
Answer:
b.
Answer: