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Unit 1 – Chapters 1 and 4
Unit 1
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Section 1.6-1.7
Section 4.1
Section 4.2
Section 4.3
Section 4.4
Section 4.5
Section 4.6-4.7
Review Ch. 4
Warm-Up – 1.6-1.7
Lesson 1.6, For use with pages 35-41
3. Gabe spent $4 more than twice as much as Casey at
a store. If Casey spent $6, how much did Gabe
spend?
ANSWER $16
Lesson 1.7, For use with pages 42-48
1. Make a table for y = 2x + 3 with x-values of 0, 3, 6, and 9.
ANSWER
x
y
0
3
3 6 9
9 15 21
2. Write a rule for the function.
Input, x 0
Output, y 1
ANSWER
y = 3x + 1
2 4 9
7 13 28
Vocabulary – 1.6-1.7
• Domain
• Quadrants
• Set of INPUTS to a function
• 4 regions of a coordinate
plane
• Sometimes these are
considered the X variables • Function
• AKA the “independent”
variable
• Range
• Set of OUTPUTS of a
function
• Sometimes these are
considered the Y variables
• AKA the “dependent”
variable
• A numerical relationship
where ONE input has
EXACTLY ONE output
• Relation
• A set of inputs and
corresponding outputs
Notes – 1.6-1.7 – Functions – Intro.
•To be a Function
1. Each INPUT must go to EXACTLY ONE output!
2. Graph must pass the Vertical Line Test
•Is a function a relation?
• Yes
•Is a relation a function?
• Sometimes!!
• Only if it passes the two tests above!
•There are several ways to sketch graphs of equations,
but the most common is THIS
1. GET Y BY ITSELF!!!
2. Build a table with at least 3 values
3. Sketch the graph
Notes – 1.6-1.7 – Functions – Intro.
•4 Different Ways to view Functions
1. Verbal Rule – In English – Rarely used.
2. Graphs
3. Equations or Rules
4. Tables
Examples 1.6-1.7
EXAMPLE 1
Identify the domain and range of a function
The input-output table shows the cost of various
amounts of regular unleaded gas from the same
pump. Identify the domain and range of the function.
Input gallons
10
Output dollars
19.99 23.99
12
13
17
25.99
33.98
ANSWER
The domain is the set of inputs: 10, 12, 13,
and 17. The range is the set of outputs:
19.99, 23.99, 25.99, and 33.98.
GUIDED PRACTICE
1.
for Example 1
Identify the domain and range of the function.
Input
Output
0
5
1
2
2
2
4
1
SOLUTION
The domain is the set of inputs: 0, 1, 2, and 4
The range is the set of outputs: 1, 2, and 5
EXAMPLE 2
Identify a function
Tell whether the pairing is a function.
a.
The pairing is not a function because the input 0 is
paired with both 2 and 3.
EXAMPLE 2
Identify a function
b.
Input
Output
0
0
1
2
4
8
6
12
The pairing is a function because each input is paired
with exactly one output.
for Example 2
GUIDED PRACTICE
Tell whether the pairing is a function.
2.
Input
Output
SOLUTION
3
1
3
6
9
12
6
2
9
2
12
1
1
2
2
1
The pairing is a function because each input is paired
with exactly one output.
for Example 2
GUIDED PRACTICE
Tell whether the pairing is a function.
3.
Input
Output
SOLUTION
2
0
2
4
7
2
1
4
2
7
3
0
1
2
3
The pairing is not a function because each input is not
paired with exactly one output. IT IS A RELATION,
THOUGH!!
EXAMPLE 3
Make a table for a function
The domain of the function y = 2x is 0, 2, 5, 7, and 8.
Make a table for the function, then identify the range
of the function.
SOLUTION
x
y = 2x
0
2
5
7
2 0 = 0 2 2 = 4 2 5 = 10 2 7 = 14
The range of the function is 0, 4, 10, 14, and 16.
8
2 8 = 16
EXAMPLE 4
Write a function rule
Write a rule for the function.
Input
0
1
4
6
10
Output
2
3
6
8
12
SOLUTION
Let x be the input, or independent variable, and let y
be the output, or dependent variable. Notice that each
output is 2 more than the corresponding input. So, a
rule for the function is y = x + 2.
GUIDED PRACTICE
4.
for Examples 3,4 and 5
Make a table for the function y – x = 5 with domain
10, 12, 15, 18, and 29. Then identify the range of the
function. HINT: GET Y BY ITSELF FIRST!!
SOLUTION
x
10
12
15
18
29
y- x = 5
10 – 5 =5
12 – 5 =7
15 – 5 =10
18 – 5 =13
18 – 29 =24
The range of the function is 5,7,10,13 and 24.
GUIDED PRACTICE
5.
for Examples 3,4 and 5
Write a rule for the function. Identify the domain
and the range.
Time (hours)
1
2
3
4
Pay (dollars)
8
16
24
32
SOLUTION
Let x be the input ,or independent variable and let y be
the output, or dependent variable. Notice that each
output is 8 times more than corresponding input .
So as a rule of function y = 8x;
domain 1, 2, 3 and 4;
range 8, 16, 24 and 32.
EXAMPLE 5
Write a function rule for a real-world situation
Concert Tickets
You are buying concert tickets that
cost $15 each. You can buy up to 6
tickets. Write the amount (in dollars)
you spend as a function of the
number of tickets you buy. Identify
the independent and dependent
variables. Then identify the domain
and the range of the function.
EXAMPLE 5
Write a function rule for a real-world situation
SOLUTION
Write a verbal model. Then write a function rule. Let n
represent the number of tickets purchased and A
represent the amount spent (in dollars).
Amount
spent
(dollars)
A
=
Cost
per ticket
(dollars/ticket)
=
15
•
Tickets
purchased
(tickets)
n
So, the function rule is A = 15n. The amount spent
depends on the number of tickets bought, so n is the
independent variable and A is the dependent variable.
EXAMPLE 5
Write a function rule for a real-world situation
Because you can buy up to 6 tickets, the domain of
the function is 0, 1, 2, 3, 4, 5, and 6. Make a table to
identify the range.
Number of tickets, n
0
1
2
3
4
5
6
Amount (dollars), A
0
15
30
45
60
75
90
The range of the function is 0, 15, 30, 45, 60, 75, and 90.
EXAMPLE 1
Graph a function
1
Graph the function y = 2 x with domain 0, 2, 4, 6, and 8.
SOLUTION
STEP 1
Make an input-output table.
x
0
2
4
6
8
y
0
1
2
3
4
EXAMPLE 1
Graph a function
STEP 2
Plot a point for each ordered pair (x, y).
WHY IS THE GRAPH A SCATTER PLOT AND
NOT A LINE??
for Example 1
GUIDED PRACTICE
1.
Graph the function y = 2x - 1 with domain
1, 2, 3, 4, and 5.
SOLUTION
STEP 1
Make an input-output table.
x
1
2
3
4
5
y
1
3
5
7
9
STEP 2
Plot a point for each ordered pair (x, y).
EXAMPLE 2
Graph a function
Sat Scores
The table shows the average scores on the
mathematics section of the Scholastic Aptitude Test
(SAT) in the United States from 1997 to 2003 as a
function of the time t in years since 1997. In the table, 0
corresponds to the year 1997, 1 corresponds to 1998,
and so on. Graph the function.
Years since
1997, t
0
1
2
3
4
5
6
Average
score, s
511
512
511
514
514
516
519
EXAMPLE 2
Graph a function
STEP 2
Plot the points
Does this graph pass the straight line test?
Is it a function??
EXAMPLE 2
Graph a function
Use the Vertical Line Test to determine if the graphs are
Functions.
Function
Function
NOT a
Function
Warm-Up – 4.1
Prerequisite Skills
SKILLS CHECK
Write the equation so that y is a function of x.
3
9. 6x + 4y = 16
y=–
x+4
ANSWER
2
10.
x + 2y = 5
11. –12x - y = –12
ANSWER
ANSWER
y=–
1
5
x+
2
2
y = -12x + 12
Prerequisite Skills
SKILLS CHECK
Graph the function on a coordinate plane and
give the Input/ Output table.
3.
y = x + 6; when x = 0, 2, 4, 6, and 8
ANSWER
Input
Output
0
6
2
8
4
10
6
12
8
14
EXAMPLE 2
Graph a function
Use the Vertical Line Test to determine if the graphs are
Functions.
Function
Function
NOT a
Function
Vocabulary – 4.1
• Coordinate Plane
• Y-coordinate
• Two dimensional plane
• The VERTICAL
used to graph ordered pairs component of an
of numbers (x,y)
ordered pair
• X-coordinate
• The HORIZONTAL
component of an ordered
pair
• Sometimes called the
“abscissa”
• Sometimes called the
“ordinate”
Notes – 4.1 – Plot Pts. and Graphs
• To plot points, move along the X axis first, and then
the Y axis
• You have to run before you jump (or drop!).
• Domain = Inputs
• Range = Outputs
•Usually in Table
format
•Quadrants labeled
With Roman Numerals
Examples 4.1
GUIDED PRACTICE
1.
for Example 1
Use the coordinate plane in Example 1 to give
the coordinates of points C, D, and E.
SOLUTION
C. C = (0,2)
D = (3,1)
E = (-2,-3)
GUIDED PRACTICE
2.
for Example 1
What is the y-coordinate of any point on the x-axis?
SOLUTION
y-coordinate of any point on the x-axis is 0
EXAMPLE 3
Graph a function
Graph the function y = 2x – 1 with domain – 2, – 1, 0, 1,
and 2. Then identify the range of the function.
SOLUTION
STEP 1
Make a table by substituting the domain values into
the function.
EXAMPLE 3
Graph a function
STEP 2
List the ordered pairs: (– 2, – 5),(– 1, – 3),
(0, – 1), (1, 1), (2, 3).Then graph the
function.
Identify the range. The range consists of the y-values
from the table: – 5, – 3, – 1, 1, and 3.
GUIDED PRACTICE
for Examples 2 and 3
1
7. Graph the function y = – 3 x + 2 with domain – 6, –3,
0, 3, and 6. Then identify the range of the function.
STEP 1
Make a table by substituting the domain values into
the function.
GUIDED PRACTICE
x
–6
–3
0
3
6
for Examples 2 and 3
y=– 1 x+2
3
y = – 1 (– 6) + 2 = 4
3
y = – 1 (– 3) + 2 = 3
3
y = – 1 (0) + 2 = 2
3
y = – 1 (3) + 2 = 1
3
y = – 1 (6) + 2 = 0
3
GUIDED PRACTICE
for Examples 2 and 3
STEP 2
List the ordered pairs: (– 6, 4),(– 3, 3), (0, 2), (3, 1), (6,0).
Then graph the function.
STEP 3
Identify the range. The range consists of the y-values
from the table: 0, 1, 2, 3 and 4.
EXAMPLE 4
Graph a function represented by a table
VOTING
In 1920 the ratification of the 19th
amendment to the United States
Constitution gave women the right to
vote. The table shows the number (to the
nearest million) of votes cast in
presidential elections both before and
since women were able to vote.
EXAMPLE 4
Graph a function represented by a table
Years before or
since 1920
– 12
–8
–4
0
4
8
12
Votes (millions)
15
15
19
27
29
37
40
a.
Explain how you know that the table represents a
function.
b.
Graph the function represented by the table.
c.
Describe any trend in the number of votes cast.
EXAMPLE 4
Graph a function represented by a table
SOLUTION
a. The table represents a function because
each input has exactly one output.
b. To graph the function, let x be the
number of years before or since 1920.
Let y be the number of votes cast (in
millions).
The graph of the function is shown.
EXAMPLE 4
Graph a function represented by a table
SOLUTION
c. In the three election years before
1920,the number of votes cast was
less than 20 million. In 1920, the
number of votes cast was greater
than 20 million. The number of votes
cast continued to increase in the
three election years since 1920.
Warm-Up – 4.2
You may use a
calculator on every
assignment from this
point forward unless
otherwise told not to!
Lesson 4.2, For use with pages 215-222
1. Graph y = –x – 2 with domain –2, –1, 0, 1, and 2.
ANSWER
Lesson 4.2, For use with pages 215-222
Rewrite the equation so y is a function of x.
2. 3x + 4y = 16
ANSWER
y=– 3 x+4
4
3. –6x – 2y = –12
ANSWER
y = –3x + 6
Vocabulary – 4.2
• Linear Equation
– The graph of the solutions to the function form a
STRAIGHT LINE!
Notes – 4.2 – Graph Linear Equations
• Standard Form of a Linear Equation looks like this:
•Ax + By = C, where A, B, and C are real numbers
and A and B are not both = 0
• There are several ways to sketch graphs of linear
equations, but the most common is THIS
1. GET Y BY ITSELF!!!
2. Build a table with at least 3 values (negative #,
positive #, and zero)
3. Sketch the graph
•The graphs of y = constant and x = constant are
special cases of linear equations.
•We’ll check those out in a minute!
Examples 4.2
EXAMPLE 1
Standardized Test Practice
Which ordered pair is a solution of 3x – y = 7?
A (3, 4)
B (1, –4)
C (5, –3)
D (–1, –2)
SOLUTION
Check whether each ordered pair is a solution of the
equation.
Test (3, 4):
3x – y = 7
?
3(3) – 4 = 7
5=7
Write original equation.
Substitute 3 for x and 4 for y.
Simplify.
EXAMPLE 1
Test (1, – 4):
Standardized Test Practice
3x – y = 7
?
3(1) – (– 4) = 7
7 =7
Write original equation.
Substitute 1 for x and – 4 for y.
Simplify.
So, (3, 4) is not a solution, but (1, – 4) is a solution of
3x – y = 7.
ANSWER
The correct answer is B. A
B
C
D
EXAMPLE 2
Graph an equation
Graph the equation – 2x + y = – 3.
SOLUTION
STEP 1
Solve the equation for y.
– 2x + y = – 3
y = 2x – 3
STEP 2
Make a table by choosing a few values for x and
finding the values of y.
x
–2
–1
0
1
2
y
–7
–5
–3
–1
1
EXAMPLE 2
Graph an equation
STEP 3
Plot the points. Notice that the points appear to lie on
a line.
EXAMPLE 3
Graph y = b and x = a
Graph (a) y = 2 and (b) x = – 1.
y=2
x = -1
GUIDED PRACTICE
for Examples 2 and 3
Graph the equation
2.
y + 3x = – 2
SOLUTION
STEP 1
Solve the equation for y.
y + 3x = – 2
y = – 3x – 2
GUIDED PRACTICE
for Examples 2 and 3
STEP 2
Make a table by choosing a few values for x and
finding the values of y.
x
y
–2
4
–1
1
0
–2
1
–5
2
–8
STEP 3
Plot the points. Notice that the points appear to lie on
a line.
STEP 4
Connect the points by drawing a line through them.
Use arrows to indicate that the graph goes on without
end.
GUIDED PRACTICE
3.
for Examples 2 and 3
y = 2.5
SOLUTION
For every value of x, the value of y is 2.5. The graph of
the equation y = 2.5 is a horizontal line 2.5 units above
the x-axis.
4.
x=–4
SOLUTION
For every value of y, the value of x is – 4. The graph of
the equation x = – 4 is a vertical line 4 units to the left
of the y-axis.
EXAMPLE 4
Graph a linear function
1
–
Graph the function y = 2 x + 4 with domain x > 0.
–
Then identify the range of the function.
SOLUTION
STEP 1
Make a table.
x
0
2
4
6
8
y
4
3
2
1
0
EXAMPLE 4
Graph a linear function
STEP 2
Plot the points.
STEP 3
Connect the points with a ray because the domain is
restricted.
STEP 4
Identify the range. From the graph, you can see that all
points have a y-coordinate of 4 or less, so the range of
the function is y ≤ 4.
GUIDED PRACTICE
for Example 4
5. Graph the function y = – 3x + 1 with domain x <– 0.
Then identify the range of the function.
SOLUTION
STEP 1
Make a table.
x
0
y
1
–1 –2 –3 –4
4
7
10 13
GUIDED PRACTICE
for Example 4
STEP 2
Plot the points.
STEP 3
Connect the points with a ray because the domain is
restricted.
STEP 4
Identify the range. From the graph, you can see that all
points have a y-coordinate of 1 or more, so the range of
the function is y >
– 1.
Warm-Up – 4.3
Daily Homework Quiz
1.
Graph y + 2x = 4
ANSWER
For use after Lesson 4.2
Daily Homework Quiz
For use after Lesson 4.2
2. The distance in miles an elephant walks in t
hours is given by d = 5t. The elephant walks
for 2.5 hours. Graph the function and
identify its domain and range.
ANSWER
domain: 0 <
– 2.5
–t<
range: 0 <
– d –< 12.5
Warm-up
1) Graph 5 points on the X-axis and label them.
What conclusion can you make about every point on the X-axis?
2) Graph 5 points on the Y-axis and label them.
What conclusion can you make about every point on the Y-axis?
Vocabulary – 4.3
• X-intercept
• Where a graph crosses the X axis
• Y-intercept
• Where a graph crosses the Y axis
Notes – 4.3 – Graph using Intercepts.
•The primary reason to use the Standard Form of a
linear equation is b/c it does make finding the x and y
intercepts VERY easy!
•To find the X-intercept of a function
• Set Y=0 and solve for X
• To find the Y-intercept of a function
• Set X=0 and solve for Y
•Since you only need two points to make a line
•Graph the X and Y intercepts and connect them!
Examples 4.3
EXAMPLE 1
Find the intercepts of the graph of an equation
Find the x-intercept and the y-intercept of the
graph of 2x + 7y = 28.
SOLUTION
To find the x-intercept, substitute 0 for y and solve for x.
2x + 7y = 28
2x + 7(0) = 28
28
x = 2 = 14
Write original equation.
Substitute 0 for y.
Solve for x.
EXAMPLE 1
Find the intercepts of the graph of an equation
To find the y-intercept, substitute 0 for x and solve for y.
2x +7y = 28
2(0) + 7y = 28
28
y= 7 =4
Write original equation.
Substitute 0 for x.
Solve for y.
ANSWER
The x-intercept is (14,0). The y-intercept is (0,4).
GUIDED PRACTICE
for Example 1
Find the x-intercept and the y-intercept of the
graph of the equation.
1. 3x + 2y = 6
SOLUTION
To find the x-intercept, substitute 0 for y and solve for x.
3x + 2y = 6
3x + 2(0) = 6
x=2
Write original equation.
Substitute 0 for y.
Solve for x.
EXAMPLE
1
for Example
1 graph of an equation
Find the intercepts
of the
GUIDED PRACTICE
To find the y-intercept, substitute 0 for x and solve for y.
3x +2y = 6
3(0) + 2y = 6
y =3
Write original equation.
Substitute 0 for x.
Solve for y.
ANSWER
The x-intercept is (2,0). The y-intercept is (0,3).
EXAMPLE 2
Use intercepts to graph an equation
Graph the equation x + 2y = 4.
SOLUTION
STEP 1
Find the intercepts.
x + 2y = 4
x + 2y = 4
x + 2(0) = 4
0 + 2y = 4
x = 4  x-intercept
X intercept = (4,0)
y = 2  y-intercept
Y intercept = (0,2)
EXAMPLE 2
Use intercepts to graph an equation
STEP 2
Plot points. The x-intercept is 4, so
plot the point (4, 0). The y- intercept
is 2, so plot the point (0, 2). Draw a
line through the points.
EXAMPLE 4
Solve a multi-step problem
EVENT PLANNING
You are helping to plan an awards banquet for your
school, and you need to rent tables to seat 180 people.
Tables come in two sizes. Small tables seat 4 people,
and large tables seat 6 people. This situation can be
modeled by the equation.
4x + 6y = 180
where x is the number of small tables and y is the
number of large tables.
•
Find the intercepts of the graph of the equation.
EXAMPLE 4
Solve a multi-step problem
•
Graph the equation.
•
Give four possibilities for the number of each
size table you could rent.
SOLUTION
STEP 1
Find the intercepts.
4x + 6y = 180
4x + 6(0) = 180
x = 45  x-intercept
4x + 6y = 180
4(0) + 6y = 180
y = 30  y-intercept
EXAMPLE 4
Solve a multi-step problem
STEP 2
Graph the equation.
The x-intercept is 45, so plot the
point (45, 0).The y-intercept is 30,
so plot the point (0, 30).
Since x and y both represent
numbers of tables, neither x nor y
can be negative. So, instead of
drawing a line, draw the part of the
line that is in Quadrant I.
EXAMPLE 4
Solve a multi-step problem
STEP 3
Find the number of tables. For this problem, only
whole-number values of x and y make sense. You can
see that the line passes through the points
(0, 30),(15,20),(30, 10), and (45, 0).
EXAMPLE 4
Solve a multi-step problem
So, four possible combinations of tables that will seat
180 people are: 0 small and 30 large, 15 small and 20
large, 30 small and 10 large,and 45 small and 0 large.
Warm-Up – 4.4
1) Do pages 10-13 from the
“Classified” ads packet as a
group.
2) You have ~15 minutes to
work on this.
Vocabulary – 4.4
• Rate of Change
• Ratio of How much something changed over
how long did it take to change.
• Slope
– The STEEPNESS of a line
– Same thing as UNIT RATE!!!!
– Same thing as Rate of Change!!!!
– HOW FAST SOMETHING IS CHANGING!!!
• Rise
– Vertical or UP/DOWN change
– Change in Y’s
• Run
– Horizontal or LEFT/RIGHT change
– Change in X’s
Notes – 4.4–Slope and Rate of Change
NOTES

Slope = RISE
RUN


4 Kinds of Slope
1.
Positive = Slants UP
2.
Negative = Slants DOWN
3.
Zero = Horizontal line
4.
No Slope = Vertical line
Finding slope with a Graph
1.
Draw a right triangle connecting the points
2.
Calculate RISE and RUN
3.
Use Slope = RISE/RUN
Notes – Continued
NOTES - CONTINUED

Finding slope with a Table

Slope = Change in Y = How much change = 2ND – 1ST
Change in X = How long did it take = 2ND – 1ST


SAME as UNIT RATE and RATE OF CHANGE!!!

Pay attention to positives/negatives!!
Finding slope Using Coordinates

Variable for slope is usually m.
1.
Slope = m = Y2 – Y1
=
RISE
X2 – X1 = RUN
2.
Plug in what you know and solve for what you don’t!
Examples 4.4
EXAMPLE 2
Find a negative slope
Find the slope of the line shown.
Let (x1, y1) = (3, 5) and (x2, y2) = (6, –1).
y 2 – y1
Write formula for slope.
m=
x 2 – x1
–1 – 5
Substitute.
=
6–3
–6
–2 Simplify.
=
3 =
EXAMPLE 3
Find the slope of a horizontal line
Find the slope of the line shown.
Let (x1, y1) = (– 2, 4) and (x2, y2) = (4, 4).
y 2 – y1
m=
x 2 – x1
4– 4
=
4 – (– 2)
0
=
= 0
6
Write formula for slope.
Substitute.
Simplify.
EXAMPLE 4
Find the slope of a vertical line
Find the slope of the line shown.
Let (x1, y1) = (3, 5) and (x2, y2) = (3, 1).
y 2 – y1
m=
x 2 – x1
1– 5
=
3–3
–4
=
0
Write formula for slope.
Substitute.
Division by zero is undefined.
ANSWER
Because division by zero is undefined, the slope of a
vertical line is undefined.
from 2,
a graph
EXAMPLE
2 Write an equation
for Examples
3 and 4
GUIDED PRACTICE
Find the slope of the line that passes through the
points.
4.
(5, 2) and (5, – 2)
SOLUTION
Let (x1, y1) = (5, 2) and (x2, y2) = (5, –2).
y2 – y1
m=
x2 – x1
–2 – 2
=
5–5
–4
=
0
ANSWER
Write formula for slope.
Substitute.
Division by zero is undefined.
The slope is undefined.
EXAMPLE 5
Find a rate of change
INTERNET CAFE
The table shows the cost of using a
computer at an Internet cafe for a
given amount of time. Find the rate
of change in cost with respect to
time.
Time(hours)
2
4
6
Cost (dollars)
7
14
21
EXAMPLE 5
Find a rate of change
SOLUTION
change in cost
Rate of change =
change in time
14 –7
7
=
=
= 3.5
2
4–2
ANSWER
The rate of change in cost is $3.50 per hour.
GUIDED PRACTICE
7.
for Example 5
EXERCISE
The table shows the distance a person walks for
exercise. Find the rate of change in distance with
respect to time.
SOLUTION
Time(minute)
30
60
90
Distance
(miles)
1.5
3
4.5
EXAMPLE
5
for of
Example
5
Find a rate
change
GUIDED PRACTICE
change in distance
Rate of change =
change in time
3 – 1.5
= 0.05
=
60 – 30
ANSWER
The rate of change in cost is $0.05 mile/minute.
EXAMPLE 6 Use a graph to find and compare rates of change
COMMUNITY THEATER
A community theater performed a play each Saturday
evening for 10 consecutive weeks. The graph shows
the attendance for the performances in weeks 1, 4, 6,
and 10. Describe the rates of change in attendance
with respect to time.
EXAMPLE 6 Use a graph to find and compare rates of change
SOLUTION
Find the rates of change using the slope formula.
Weeks 1–4:
Weeks 4–6:
Weeks 6–10:
108
232 – 124
= 36 people per week
=
4–1
3
204 – 232 = – 28 = – 14 people per week
6–4
2
– 132
72 – 204
=
= – 33 people per week
10 – 6
4
ANSWER
Attendance increased during the early weeks of
performing the play. Then attendance decreased,
slowly at first, then more rapidly.
EXAMPLE 7
Interpret a graph
COMMUTING TO SCHOOL
A student commutes from home to school by walking
and by riding a bus. Describe the student’s commute
in words.
EXAMPLE 7
Interpret a graph
SOLUTION
The first segment of the graph is not very steep, so
the student is not traveling very far with respect to
time. The student must be walking. The second
segment has a zero slope, so the student must not be
moving. He or she is waiting for the bus. The last
segment is steep, so the student is traveling far with
respect to time. The student must be riding the bus.
Warm-Up – 4.5
Lesson 4.5, For use with pages 243-250
1. Rewrite 6x + 2y = 8 so y is a function of x.
ANSWER
y = –3x + 4
2. Find the slope of the line that passes through (–5, 6)
and (0, 8).
ANSWER
2
5
Lesson 4.5, For use with pages 243-250
3. Find the intercepts of the graph of the function. (USE
THE COORDINATES OF THE POINTS!)
200x + 100y = – 600.
ANSWER
y-intercept: (0,– 6), x-intercept: (- 3,0)
4a Find the slope of -2x + y = 1.
4b Find the y-intercept of this equation as well.
HINT 1: GET Y BY ITSELF.
HINT 2: BUILD A TABLE!
ANSWER
Slope = 2 and y-intercept is 1.
Do these numbers look familiar???????
Vocabulary – 4.5
• Parallel Lines
– Lines that never intersect
– Lines that have the SAME SLOPE!
• slope-intercept form
– Linear equation where y = mx + b
– m = slope of the line
– b = y-intercept
Notes – 4.5 – Slope-Intercept Form
Slope Intercept Form of an Equation –




y = mx + b
To use the slope intercept form
1.
Solve the equation so that Y is by itself
2.
The coefficient of X is the slope.
3.
The constant number is the Y intercept.
To graph a function using the slope intercept form
1.
Graph the Y intercept
2.
Use the slope = rise/run to find the next point
3.
Graph the second point and connect the two points
If two lines are parallel, their slopes are ?????

Identical!
BrainPop:
Slope and Intercept
Examples 4.5
Examples 4.5 – Using the Graphing Calculator
1. Open up the graphing application.
2. Graph the equation f1(x) = 2x+1 and press the
Graph button. What happens?
3. Graph f2(x) = 2x + 3 What happens?
4. Press the Table button (you have to find it first!).
What happens?
5. On the menu, press the Window/Zoom option
button and choose 5- Zoom standard. What
happens?
6. Graph the function f3(x)= x^2. Look familiar?
7. Clear all the functions by pressing Actions
Delete All
EXAMPLE 1
Identify slope and y-intercept
Identify the slope and y-intercept of the line with the
given equation.
1. y = 3x + 4
2. 3x + y = 2
SOLUTION
a. The equation is in the form y = m x + b. So, the slope
of the line is 3, and the y-intercept is 4.
b. Rewrite the equation in slope-intercept form by
solving for y.
EXAMPLE 1
Identify slope and y-intercept
3x + y = 2
y = – 3x + 2
Write original equation.
Subtract 3x from each side.
ANSWER
The line has a slope of – 3 and a y-intercept of 2.
EXAMPLE
1
for Example
1
Identify slope
and y-intercept
GUIDED PRACTICE
Identify the slope and y-intercept of the line with the
given equation.
1. y = 5x – 3
SOLUTION
The equation is in the form y = mx + b. So, the slope
of the line is 5, and the y-intercept is –3.
EXAMPLE
1
for Example
1
Identify slope
and y-intercept
GUIDED PRACTICE
Identify the slope and y-intercept of the line with the
given equation.
2. 3x – 3y = 12
EXAMPLE
1
for Example
1
Identify slope
and y-intercept
GUIDED PRACTICE
SOLUTION
Rewrite the equation in slope-intercept form by
solving for y.
Write original equation.
3x – 3y = 12
Rewrite original equation.
3x – 12 = 3y
Divide 3 by equation.
y = 3x + 12
3
Simplify.
y =x – 4
ANSWER
The line has a slope of –1 and a y-intercept of –4.
EXAMPLE
1
for Example
1
Identify slope
and y-intercept
GUIDED PRACTICE
Identify the slope and y-intercept of the line with the
given equation.
3. x + 4y = 6
EXAMPLE
1
for Example
1
Identify slope
and y-intercept
GUIDED PRACTICE
SOLUTION
Rewrite the equation in slope-intercept form by
solving for y.
Write original equation.
x + 4y = 6
4y = – x + 6
y= – x + 6
4
–x
6
= 4 + 4
Rewrite original equation.
Divide 3 by equation.
Simplify.
ANSWER
–
The line has a slope of 1 and a y-intercept of 1 1 .
4
2
EXAMPLE 2 Graph an equation using slope-intercept form
Graph the equation 2x + y = 3.
SOLUTION
STEP 1
Rewrite the equation in slope-intercept form.
y = – 2x + 3
EXAMPLE 2 Graph an equation using slope-intercept form
STEP 2
Identify the slope and the y-intercept.
m= – 2
and
b =3
STEP 3
Plot the point that corresponds to the y-intercept,(0, 3).
STEP 4
Use the slope to locate a second point on the line.
Draw a line through the two points.
EXAMPLE 3
Change slopes of lines
ESCALATORS
To get from one floor to another at a
library, you can take either the stairs
or the escalator. You can climb stairs
at a rate of 1.75 feet per second, and
the escalator rises at a rate of 2 feet
per second. You have to travel a
vertical distance of 28 feet. The
equations model the vertical distance
d (in feet) you have left to travel after t
seconds.
Stairs: d = – 1.75t + 28
Escalator: d = – 2t + 28
EXAMPLE 3
Change slopes of lines
a.
Graph the equations in the same coordinate plane.
b.
How much time do you save by taking the escalator?
SOLUTION
a. Draw the graph of d = – 1.75t + 28
using the fact that the d-intercept
is 28 and the slope is – 1.75.
Similarly, draw the graph of
d = – 2t + 28. The graphs make
sense only in the first quadrant.
EXAMPLE 3
b.
Change slopes of lines
The equation d = – 1.75t + 28 has a t-intercept of 16.
The equation d = – 2t + 28 has a t-intercept of 14. So,
you save 16 – 14 = 2 seconds by taking the
escalator.
EXAMPLE
2 Graph an equation
for Examples
2 and
3
using
slope-intercept
form
GUIDED PRACTICE
4.
Graph the equation y = – 2x + 5.
SOLUTION
STEP 1
Identify the slope and the y- intercept.
m= – 2
and
b =5
STEP 2
Plot the point that corresponds to the y-intercept, (0, 5).
STEP 3
Use the slope to locate a second point on the line.
Draw a line through the two points.
EXAMPLE 5
Identify parallel lines
Determine which of the lines are
parallel.
Find the slope of each line.
–1– 0
–1
1
Line a: m =
= –3 =
–1– 2
3
Line b: m = – 3 – (–1 )
0 – 5
– 5 – (–3)
Line c: m = – 2 – 4
–2
2
= –5 =
5
–2
1
= –6 = 3
ANSWER
Line a and line c have the same slope, so they are
parallel.
EXAMPLE
5
for Examples
Identify parallel
lines 4 and 5
GUIDED PRACTICE
7.
Determine which lines are parallel: line a through
(-1, 2) and (3, 4); line b through (3, 4) and (5, 8); line
c through (-9, -2) and (-1, 2).
SOLUTION
Find the slope of each line.
4– 2
2
1
Line a: m =
3 –(1) = 3+1 = 2
8– 4
–4
Line b: m = 5 – 3
= –2
= 2
2 – (–2)
2+2
Line c: m = – 1 – (– 9) =–1+ 9
4
= 8
1
= 2
Warm-Up – 4.6 and 4.7
Lesson 4.6, For use with pages 253-259
Write the equation in slope intercept form and
sketch the graph
1. 4x – y = –8
ANSWER
y=4x+8
2. -9x - 3y = 21
ANSWER
y = -3x - 7
Lesson 4.6, For use with pages 253-259
Write the equation and sketch the graph
1. Slope = 1 and y-int = 2
ANSWER
Y=x+2
Lesson 4.6, For use with pages 253-259
Find the unit rate.
3. You are traveling by bus. After 4.5 hours, the bus has
traveled 234 miles. Use the formula d = rt where d is
distance, r is a rate, and t is time to find the average
rate of speed of the bus.
ANSWER
52 mi/h
Lesson 4.6, For use with pages 253-259
Sketch the graph
1. y = 4x
ANSWER
Vocabulary – 4.6-4.7
• Direct Variation
– Linear equation where y = kx
• Constant of Variation
– In a Direct Variation, the letter
k is the constant of variation
– It’s the unit rate, the rate of
change and …
– SAME AS SLOPE!!
• Function Notation
– Different way of writing
functions
– F(x) means the “the function
of x”
• Family of Functions
– A group of functions
with similar
characteristics (e.g.
their graphs are all
linear)
• Parent Linear
Function
– Simplest form of a
family of functions
– F(x) = x is the parent
linear function
Notes – 4.6-4.7 – Direct Variations and
Graphing Linear Functions
A Direct variation has the form
 y = kx
 k = the constant of variation and is AKA
 THE SLOPE!
 A direct variation graph ALWAYS goes through the origin.
 A direct variation is ALWAYS PROPORTIONAL!
 There are two ways to find the constant, k
1.Find the UNIT RATE
2.If they give you a coordinate (x,y),
1. Plug in the numbers to y = kx
2. Solve for k.
Notes – 4.6-4.7 – Direct Variations
and Graphing Linear Functions – cont.
A “function” is usually written as
F(x) and we read it as “F of x”
OR y = F(x)
To evaluate functions
Plug in what you know and …..???
Examples 4.6-4.7
EXAMPLE 1
Identify direct variation equations
Tell whether the equation represents direct variation.
If so, identify the constant of variation.
a. 2x – 3y = 0
b. – x + y = 4
EXAMPLE 1
Identify direct variation equations
SOLUTION
To tell whether an equation represents direct
variation, try to rewrite the equation in the form y = ax.
2x – 3y = 0
– 3y = – 2x
2x
y= 3
Write original equation.
Subtract 2x from each side.
Simplify.
ANSWER
Because the equation 2x – 3y = 0 can be
rewritten in the form y = ax, it represents direct
variation. The constant of variation is. 2
3
EXAMPLE 1
b.
Identify direct variation equations
–x+y=4
y = x+4
Write original equation.
Add x to each side.
ANSWER
Because the equation – x + y = 4 cannot be
rewritten in the form y = ax, it does not represent
direct variation.
GUIDED PRACTICE
for Example 1
Tell whether the equation represents direct variation.
If so, identify the constant of variation.
1.
–x+y=1
GUIDED PRACTICE
for Example 1
SOLUTION
To tell whether an equation represents direct
variation, try to rewrite the equation in the form y = ax.
–x+y=1
y = x+1
Write original equation.
Add x each side.
ANSWER
Because the equation – x + y = 1 cannot be
rewritten in the form y = ax, it does not represent
direct variation.
GUIDED PRACTICE
2.
for Example 1
2x + y = 0
SOLUTION
2x + y = 0
y = – 2x
Write original equation.
Subtract 2x from each side.
ANSWER
Because the equation 2x + y = 0 can be rewritten
in the form y = ax, it represents direct variation.
The constant of variation is. – 2
GUIDED PRACTICE
3.
for Example 1
4x – 5y = 0
SOLUTION
To tell whether an equation represents direct
variation, try to rewrite the equation in the form y = ax.
4x – 5y = 0
Write original equation.
4x = 5y
Subtract Add 5y each side.
4x
Simplify.
y= 5
ANSWER
Because the equation 4x – 5y = 0 can be
rewritten in the form y = ax, it represents direct
variation. The constant of variation is. 4
5
EXAMPLE 2
Graph direct variation equations
Graph the direct variation equation.
a. y =
2
x
3
b. y = – 3x
SOLUTION
a.
Plot a point at the origin.
The slope is equal to the
constant of variation, or 2
Find and plot a second 3
point, then draw a line
through the points.
EXAMPLE 2
b.
Graph direct variation equations
Plot a point at the origin. The slope is
equal to the constant of variation, or – 3.
Find and plot a second point, then draw
a line through the points.
EXAMPLE 3
Write and use a direct variation equation
The graph of a direct variation
equation is shown.
a. Write the direct variation equation.
b. Find the value of y when x = 30.
SOLUTION
a. Because y varies directly with x, the equation
has the form y = ax. Use the fact that y = 2 when
x = – 1 to find a.
y = ax
2 = a (– 1)
–2=a
Write direct variation equation.
Substitute.
Solve for a.
EXAMPLE 3
Graph direct variation equations
ANSWER
A direct variation equation that relates x
and y is y = – 2x.
b. When x = 30, y = – 2(30) = – 60.
GUIDED PRACTICE
4.
for Examples 2 and 3
Graph the direct variation equation.
y = 2x
SOLUTION
Plot a point at the origin. The slope is equal to the
constant of variation, or Find and plot a second
point, then draw a line through the points.
GUIDED PRACTICE
for Examples 2 and 3
5. The graph of a direct variation on equation passes
through the point (4,6). Write the direct variation
equation and find the value of y when x =24.
SOLUTION
Because y varies directly with x, the equation has
the form y = ax. Use the fact that y = 6 when x = 4
to find a.
y = ax
Write direct variation equation.
6 = a (4)
Substitute.
3
6
=
2
4
Solve for a.
a=
GUIDED PRACTICE
for Examples 2 and 3
ANSWER
A direct variation equation that relates x
and y is y = 3 x when x = 24. y = 3 (24) = 36
2
2
EXAMPLE 1
Standardized Test Practice
SOLUTION
f (x)= 3x – 15
f (– 3) = 3(– 3) – 15
= 24
Write original function.
Substitute -3 for x.
Simplify.
ANSWER
The correct answer is A. A
B C
D
GUIDED PRACTICE
1.
for Example 1
Evaluate the function h(x) = – 7x when x = 7.
SOLUTION
h(x) = – 7x
Write original function.
h(7) = – 7(7)
Substitute 7 for x.
= – 49
Simplify.
EXAMPLE 2
Find an x-value
For the function f(x) = 2x – 10, find the value of x so that
f(x) = 6.
f(x) = 2x – 10
Write original function.
6 = 2x – 10
Substitute 6 for f(x).
8=x
Solve for x.
ANSWER
When x = 8, f(x) = 6.
EXAMPLE 3
Graph a function
GRAY WOLF
The gray wolf population in
central Idaho was monitored
over several years for a
project aimed at boosting the
number of wolves. The
number of wolves can be
modeled by the function f(x) =
37x + 7 where x is the number
of years since 1995. Graph the
function and identify its
domain and range.
Review – Chap. 4
Daily Homework Quiz
1.
Write the equation 15 = 5y – 4x so that y is a
function of x
ANSWER
2.
y= 4 x+3
5
Solve C = 2pr for r
ANSWER
3.
For use after Lesson 3.8
r=
C
p
2
Solve V = 1 Bh for B.
3
ANSWER
B = 3V
h
Daily Homework Quiz
For use after Lesson 3.8
4. On a round-trip bicycle trip from Santa
Barbara to Canada, Phil rode 2850 miles in
63 days. Find his average miles per day for
the trip. Use the formula d = rt where d is
distance, r is rate, and t is time.Solve for r
to find the rate in miles per day to the
nearest mile.
ANSWER
about 45 miles per day
Daily Homework Quiz
For use after Lesson 4.1
Give the coordinates of the points. Plot the points in a
coordinate plane
1. and 2.
ANSWER
A(-3,2)
ANSWER
B(0,-1)
Daily Homework Quiz
3.
A(-2,-4)
ANSWER
For use after Lesson 4.1
Daily Homework Quiz
For use after Lesson 4.1
4. B(3,0)
ANSWER
5.
1
Graph y = 2 x – 1 with domain – 4, – 2, 0, 2, 4. Then
identify the range
ANSWER
– 3, – 2, – 1, 0, 1
Daily Homework Quiz
1.
Graph y + 2x = 4
ANSWER
For use after Lesson 4.2
Daily Homework Quiz
For use after Lesson 4.2
2. The distance in miles an elephant walks in t
hours is given by d = 5t. The elephant walks
for 2.5 hours. Graph the function and
identify its domain and range.
ANSWER
domain: 0 <
– 2.5
–t<
range: 0 <
– d –< 12.5
Daily Homework Quiz
1.
For use after Lesson 4.3
Find the x-intercept and the y-intercept of the
graph of 3x – y = 3
ANSWER
x-int: 1, y-int: – 3
Daily Homework Quiz
2.
For use after Lesson 4.3
A recycling company pays $1 per used ink jet
cartridge and $2 per used cartridge. The
company paid a customer $14.This situation is
given by x + 2y = 14 where x is the number of
inkjet cartridges and y the number of laser
cartridges. Use intercepts to graph the equation.
Give four possibilities for the number of each
type of cartridge that could have been recycled.
ANSWER
(0,7),(6,4),(10,2),(14,0)
Daily Homework Quiz
For use after Lesson 4.4
Find the slope of the line that passes through the points
1. (12, – 1) and (– 3, – 1)
ANSWER
2.
0
(–2, 6) and (4, –3)
ANSWER
– 3
2
Daily Homework Quiz
3.
For use after Lesson 7.2
The graph shows the
ticket sales for a school
dance on day 1, day 3,
day 6, and day 9,of ticket
sales. Describe the rates
of change in ticket sales
with respect to time.
ANSWER
Ticket sales grew moderately, declined slightly, and
then had another moderate rate of increase.
Daily Homework Quiz
1.
Identify the slope and y-intercept of the line
2x + 4y = –16.
ANSWER
2.
For use after Lesson 4.5
y =
2 x
+1
3
ANSWER
1
Slope: –
,y-intercept: –4
2
Daily Homework Quiz
3.
For use after Lesson 7.2
Determine which of the lines are parallel.
ANSWER
Lines a and c
Daily Homework Quiz
1.
Identify the slope and y-intercept of the line
2x + 4y = –16.
ANSWER
2.
For use after Lesson 4.5
y =
2 x
+1
3
ANSWER
1
Slope: –
,y-intercept: –4
2
Daily Homework Quiz
3.
For use after Lesson 7.2
Determine which of the lines are parallel.
ANSWER
Lines a and c
Daily Homework Quiz
For use after Lesson 4.6
Tell whether the equation represents direct variation.
If so, identify the constant of variation.
1.
5x – 6y = 2
ANSWER
2.
no
x+y = 0
ANSWER
yes, – 1
Daily Homework Quiz
3.
For use after Lesson 4.6
The number p of parts a machine produces
varies directly with the time t (in minutes) the
machine is in operation. The machine
produces 84 parts in 14 minutes. Write a
direct variation equation that relates t and p.
how many parts does the machine produce in
25 minutes?
ANSWER
p = 6t; 150 parts
Daily Homework Quiz
For use after Lesson 4.7
1. Evaluate f (x) = 8x – 4 when x = –3, 0, and 2.
ANSWER
–28, –4, 12
2. Find the value of x so g (x) = –2x+ 1 has the value –3.
ANSWER
2
Daily Homework Quiz
For use after Lesson 4.7
3. A stable charges $25 for feed and $50
per day to stable horses. The cost is
given by f(x) = 50x + 25. Recently, the
stable raised its fee for food to $50.
The new fee is given by g(x) = 50x +
50.Graph the functions and then
compare the two graphs.
ANSWER
The graphs have the same
slope.The y-intercept of g is 25
units greater than that of f, so g
is a vertical translation.
Warm-Up – X.X
Vocabulary – X.X
• Holder
• Holder 2
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• Holder 4
Notes – X.X – LESSON TITLE.
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Examples X.X