Transcript No Slide Title

Sequences
LESSON 11-1
Course 3
Problem of the Day
Wendell, Karla, Rosalie, and Lynne are each traveling to a different
city this summer. The cities are Paris, Sydney, Tokyo, and Mexico
City. Karla is staying in North America and Lynne is going to Europe.
Rosalie can hardly wait to see kangaroos in their natural habitat. Who
is going to what city?
Wendell to Tokyo, Karla to Mexico City, Rosalie to Sydney,
Lynne to Paris
Lesson
Main
Lesson
11-1
Feature
Sequences
LESSON 11-1
Course 3
Check Skills You’ll Need
(For help, go to Lesson 1-1.)
1. Vocabulary Review How do you evaluate an expression?
Write an algebraic expression for each word phrase.
2. 7 more than a number.
3. 5 times a number.
4. the number of eggs in d dozen.
Check Skills You’ll Need
Lesson
Main
Lesson
11-1
Feature
Sequences
LESSON 11-1
Course 3
Check Skills You’ll Need
Solutions
1. Replace each variable with a number and then simplify.
2. n + 7
3. 5n
4. 12d
Lesson
Main
Lesson
11-1
Feature
Sequences
LESSON 11-1
Course 3
Additional Examples
Find the next three terms in the sequence 3, 10, 17, 24, . . . .
You find each term by adding 7 to the previous term.
The next three terms are 31, 38, and 45.
Quick Check
Lesson
Main
Lesson
11-1
Feature
Sequences
LESSON 11-1
Course 3
Additional Examples
Find the first four terms of the sequence represented by the
expression 3 + 6n.
Term number (n)
Term of
Sequence
1
2
3
4
3+6•1
=9
3+6•2
= 15
3+6•3
= 21
3+6•4
= 27
Place the results in a series as 9, 15, 21, 27.
Quick Check
Lesson
Main
Lesson
11-1
Feature
Sequences
LESSON 11-1
Course 3
Additional Examples
Find an algebraic expression for the sequence –3, 3, 9, 15, . . .
Use the expression to find the next term in the sequence.
The common difference is 6. To get the second term, you take –3 and
add 6 one time. To get the third term, you add 6 to –3 two times.
Words
start with –3
+
6 times one less than
the term number
Let n = the number of the term.
Expression
–3
+ 6(n – 1)
The next term is for n = 5: –3 + 6(5 – 1) = –3 + 6(4) = 21.
Lesson
Main
Lesson
11-1
Quick Check
Feature
Sequences
LESSON 11-1
Course 3
Additional Examples
A scientist isolates 10 cells in a dish. The next day there
are 40 cells in the dish. The day after there are 160 cells. Write the
rule for the geometric sequence and find the next three terms.
10
40
4
160
4
640
4
2,560 10,240
4
4
The common ratio is 4.
The rule is, Start with 10 and multiply by 4 repeatedly. The next
three terms are 640; 2,560; and 10,240.
Quick Check
Lesson
Main
Lesson
11-1
Feature
Sequences
LESSON 11-1
Course 3
Lesson Quiz
Write a rule for sequences in Examples 1 and 2.
1. 1, 8, 64, 512, . . .
Start with 1 and multiply by 8 repeatedly.
2. 100, 85, 70, 55, . . .
Start with 100 and add –15 repeatedly.
3. Write an algebraic expression for the arithmetic sequence
–6, –2, 2, 6, . . . .
–6 + 4(n – 1)
4. Write the first four terms of the sequence represented by
–7n + 21.
14, 7, 0, –7
Lesson
Main
Lesson
11-1
Feature
Relating Graphs to Events
LESSON 11-2
Course 3
Problem of the Day
In a certain part of a state, only the letters J, K, L, M, and N can be
used to form a 2-letter beginning for a license plate. How many
different 2-letter beginnings are possible?
25 beginnings; 20 using two different letters and 5 using the same
letter twice
Lesson
Main
Lesson
11-2
Feature
Relating Graphs to Events
LESSON 11-2
Course 3
Check Skills You’ll Need
(For help, go to Lesson 9-9.)
1. Vocabulary Review What type of data sets do line graphs best
display?
Describe a set of data that is appropriate for each graph.
2. line plot
3. bar graph
Check Skills You’ll Need
Lesson
Main
Lesson
11-2
Feature
Relating Graphs to Events
LESSON 11-2
Course 3
Check Skills You’ll Need
Solutions
1.
Line graphs best display changes over time.
2–3. Answers may vary, samples are given.
2.
Line plots best display frequency of data—for example,
displaying the number of siblings each class member has.
3.
Bar graphs compare amounts in different categories—for example,
the number of students in each grade
Lesson
Main
Lesson
11-2
Feature
Relating Graphs to Events
LESSON 11-2
Course 3
Additional Examples
Quick Check
Use the graph below.
a. How long did the trip take?
Time is shown on the x-axis. The trip lasted 1 hour, from 7:00 to 8:00.
b. What was the fastest speed?
The fastest speed was 50 mi/h. This speed was maintained between
7:40 and 7:50.
Lesson
Main
Lesson
11-2
Feature
Relating Graphs to Events
LESSON 11-2
Course 3
Additional Examples
An athlete jogs for 30 min, sprints for 5 min, and walks for
10 min. Sketch and label a graph showing his speed.
Quick Check
Lesson
Main
Lesson
11-2
Feature
Relating Graphs to Events
LESSON 11-2
Course 3
Lesson Quiz
1. Water is steadily poured into a cone-shaped vase. Sketch
and label a graph for the water level as the vase is filled.
2. The graph below describes changes in the water level between low tides at
the Bay of Fundy on the eastern coast of Canada. What is the approximate
amount of time between low tides? What is the greatest distance between
high tide and low tide?
about 12 hours; about 40 feet
Lesson
Main
Lesson
11-2
Feature
Relating Graphs to Events
LESSON 11-2
Course 3
Lesson Quiz
3. An airplane flew from Los Angeles to San Francisco in 70 min. The plane took
20 min. to reach its cruising altitude. It took 15 min. to descend into San
Francisco. Sketch and label a graph that shows the plane’s altitude during the
flight.
Lesson
Main
Lesson
11-2
Feature
Functions
LESSON 11-3
Course 3
Problem of the Day
A rectangular field is three times as long as it is wide. What are its
width and length if the perimeter is 600 yd?
width: 75 yd; length: 225 yd
Lesson
Main
Lesson
11-3
Feature
Functions
LESSON 11-3
Course 3
Check Skills You’ll Need
(For help, go to Lesson 1–1.)
1. Vocabulary Review What is the variable in the expression
3a + 7?
Evaluate each expression for v = 7.
2. 2(v – 3)
3. 7v + 4
4. 3v – 12
5. –5(15 – 2v)
Check Skills You’ll Need
Lesson
Main
Lesson
11-3
Feature
Functions
LESSON 11-3
Course 3
Check Skills You’ll Need
Solutions
1. a
2. 2(7 – 3) = 2(4) = 8
3. 7(7) + 4 = 49 + 4 = 53
4. 3(7) – 12 = 21 – 12 = 9
5. –5(15 – 2 • 7) = – 5(15 – 14) = – 5(1) = – 5
Lesson
Main
Lesson
11-3
Feature
Functions
LESSON 11-3
Course 3
Additional Examples
Complete the table for p = 4s.
Input s Output p
3
4  3 = 12
5
4  5 = 20
7
4  7 = 28
9
4  9 = 36
Quick Check
Lesson
Main
Lesson
11-3
Feature
Functions
LESSON 11-3
Course 3
Additional Examples
Use the function rule f(x) = 3x – 1. Find the output values f(5).
f(x) = 3x – 1
Write the function rule.
f(5) = 3 • 5 – 1
Substitute the input value for x.
= 15 – 1
Simplify.
= 14
Quick Check
Lesson
Main
Lesson
11-3
Feature
Functions
LESSON 11-3
Course 3
Additional Examples
Use functional notation to show the relationship between the
total number of cars and the number of tires. Identify your variables.
Words
number of cars
=
number of tires
Let t = the number of tires.
input
Let f(t) = the number of cars.
output
Function
f(t)
= t
f(t) =
4
4
t
4
Quick Check
Lesson
Main
Lesson
11-3
Feature
Functions
LESSON 11-3
Course 3
Lesson Quiz
1. A photocopy costs $.08. Use the function rule c = $0.08n. Make a
table of input/output pairs to show the cost for 5, 10, and 15
copies.
Input n
5
10
15
Output c $.40 $.80 $1.20
Evaluate each of the following for the function rule f(x) = 40 – 2x.
2. f(12)
3. f(–12)
64
16
4. Suppose peaches cost $.99 per pound. Write a function rule to
describe the relationship between the total cost C(p) and the
number of pounds of peaches p you buy.
C(p) = 0.99p
Lesson
Main
Lesson
11-3
Feature
Understanding Slope
LESSON 11-4
Course 3
Problem of the Day
Give the number of faces, edges, and vertices of a rectangular prism.
6 faces, 12 edges, and 8 vertices
Lesson
Main
Lesson
11-4
Feature
Understanding Slope
LESSON 11-4
Course 3
Check Skills You’ll Need
(For help, go to Lesson 1-3.)
1. Vocabulary Review Do the negative integers include zero?
Simplify each expression.
2. – 3 – 1
3. 10 – (– 4)
4. 1 – 7
5.
– 8 – (– 6)
Check Skills You’ll Need
Lesson
Main
Lesson
11-4
Feature
Understanding Slope
LESSON 11-4
Course 3
Check Skills You’ll Need
Solutions
1. no
2. – 3 – 1= – 3 + (– 1) = – 4
3. 10 – (– 4) = 10 + 4 = 14
4. 1 – 7 = 1 + (– 7) = – 6
5. – 8 – (– 6) = – 8 + 6 = – 2
Lesson
Main
Lesson
11-4
Feature
Understanding Slope
LESSON 11-4
Course 3
Additional Examples
Using coordinates, find the slope of the line
between P (–2, 3) and Q (–1, –1).
slope =
change in y
change in x
3 – (–1)
= –2 – (–1)
Subtract the coordinates
of point Q from the coordinates
of point P.
= 4 or –4
–1
Simplify.
Quick Check
Lesson
Main
Lesson
11-4
Feature
Understanding Slope
LESSON 11-4
Course 3
Additional Examples
Find the slope of each line. State whether the
slope is zero or undefined.
a. line k
b. line p
1 – (–3)
4
2–2
0
slope = 2 – 2 = 0
slope = –2 – 3 = –5 = 0
Division by zero is
undefined. The slope
of a vertical line is
The slope of a
horizontal line
is zero.
undefined.
Quick Check
Lesson
Main
Lesson
11-4
Feature
Understanding Slope
LESSON 11-4
Course 3
Additional Examples
Quick Check
Graph the distance-cost data below. Connect the points with a
line. Then find the rate of change.
Distance (mi)
0
100
200
300
400
500
Cost ($)
0
25
50
75
100
125
rate of
change in y
25 – 0
=
slope
=
=
Change
change in x
100 – 0
=
25
1
=
100
4
Use coordinates
of two points.
Subtract and simplify.
The amount spent on fuel increases $1 for every 4 miles driven.
Lesson
Main
Lesson
11-4
Feature
Understanding Slope
LESSON 11-4
Course 3
Lesson Quiz
1. Find the slope of line s.
2
3
2. Find the slope of line t.
undefined
3. The data in the table is linear. Find the slope.
3
4. Graph Exercise 3 data and the line.
Lesson
Main
Lesson
11-4
Feature
Graphing Linear Functions
LESSON 11-5
Course 3
Problem of the Day
Find the area of a square 15.7 mm on each side.
246.49 mm2
Lesson
Main
Lesson
11-5
Feature
Graphing Linear Functions
LESSON 11-5
Course 3
Check Skills You’ll Need
(For help, go to Lesson 11–4.)
1.
Vocabulary Review Explain how to find the slope of a line.
Find the slope of the line that passes through each pair of points.
2.
(2, 4), (–5, 10)
3.
(0, 0), (6, 0)
4.
(2, 1), (1, 2)
Check Skills You’ll Need
Lesson
Main
Lesson
11-5
Feature
Graphing Linear Functions
LESSON 11-5
Course 3
Check Skills You’ll Need
Solutions
1.
Divide the change in y by the change in x.
2.
– 6
7
3.
0
4.
–1
Lesson
Main
Lesson
11-5
Feature
Graphing Linear Functions
LESSON 11-5
Course 3
Additional Examples
Juice costs $2.19 per gallon. The total cost of g gallons is a
function of the price of a single gallon. Make a table and a graph.
You cannot buy part of a container, so the data are discrete.
Use points for each input value.
Connect the points with a dashed line.
Number of
Gallons
1
2
3
4
Total Cost
$2.19
$4.38
$6.57
$8.76
Quick Check
Lesson
Main
Lesson
11-5
Feature
Graphing Linear Functions
LESSON 11-5
Course 3
Additional Examples
Amber earns $7 per hour. Make a table to describe Amber’s
earnings (output) as a function of hours she works (input). Graph the
function.
Input (h)
0
1
2
3
4
5
Output ($)
0
7
14
21
28
35
Quick Check
Lesson
Main
Lesson
11-5
Feature
Graphing Linear Functions
LESSON 11-5
Course 3
Lesson Quiz
1. A rock climber 150 ft above the base of a cliff climbs at a rate
of 5 ft per min. Make a table to describe the climber’s height
above the cliff base as a function of elapsed time. Graph the
function.
Time (min) 0
1
2
Height (ft) 150 155 160
2. Graph the function y = –3x – 4.
Lesson
Main
Lesson
11-5
Feature
Graphing Linear Functions
LESSON 11-5
Course 3
Lesson Quiz
3. One kilogram is equivalent to about 2.2 pounds. The weight of
an object in pounds is a function of its weight in kilograms.
Make a table and graph the function.
Kilograms (kg)
Pounds (lb)
Lesson
Main
0
0
1
2.2
Lesson
11-5
2
4.4
3
6.6
Feature
Writing Rules for Linear Functions
LESSON 11-6
Course 3
Problem of the Day
At the football game, a team scored 44 points. They scored 3 points for
each field goal and 7 points for each touchdown and its successful
conversion. What combinations of field goals and touchdown might
they have made?
3 field goals and 5 touchdowns, or 10 field goals and 2 touchdowns
Lesson
Main
Lesson
11-6
Feature
Writing Rules for Linear Functions
LESSON 11-6
Course 3
Check Skills You’ll Need
(For help, go to Lesson 11–5.)
1. Vocabulary Review What is slope-intercept form?
For each function, find the slope and y-intercept.
2. y = 3x – 2
3. y = x + 5
4. y = 8x
Check Skills You’ll Need
Lesson
Main
Lesson
11-6
Feature
Writing Rules for Linear Functions
LESSON 11-6
Course 3
Check Skills You’ll Need
Solutions
1. y = mx + b, where m is the slope and b is the y-intercept
2. 3; – 2
3. 1; 5
4. 8; 0
Lesson
Main
Lesson
11-6
Feature
Writing Rules for Linear Functions
LESSON 11-6
Course 3
Additional Examples
Quick Check
A rate for Internet access is $15 per month plus $.25 per hour
of use. Write a function rule that shows how the monthly bill depends
on the number of hours used.
Words monthly bill = $15 + $.25 • number of hours
Let x = the number of hours.
Let y = the monthly bill.
Function
y
= 15 + 0.25 • x
y = 15 + 0.25x
The function rule y = 15 + 0.25x shows the monthly bill for x hours of use.
Lesson
Main
Lesson
11-6
Feature
Writing Rules for Linear Functions
LESSON 11-6
Course 3
Additional Examples
Do the data in the table below represent a linear function? If
so, write a rule for the function.
Find the changes in inputs.
Find the changes in outputs.
Change in y
Change in x
2
1
2
1
2
1
2
1
Compare the changes as ratios.
Since each ratio is the same, the function is linear. Use slope-intercept
form to write a function rule.
Lesson
Main
Lesson
11-6
Feature
Writing Rules for Linear Functions
LESSON 11-6
Course 3
Additional Examples
(continued)
According to the table, the point (0, 8) lies on the graph of the function. So
the y-intercept is 8. The slope is 2.
y = 2x + 8
Substitute 2 for m and 8 for b.
Quick Check
Lesson
Main
Lesson
11-6
Feature
Writing Rules for Linear Functions
LESSON 11-6
Course 3
Additional Examples
Quick Check
Find the equation of the line in the graph below.
Step 1 Find the slope.
Use the points (–4, 0) and (0, 8).
The y-intercept is 8.
slope =
8–0
=2
0 – (–4)
Step 2 Use the slope and one point to
write an equation.
y – y1 = m (x – x1)
y – 0 = 2 [x – (–4)]
Use the point (–4,0)
for (x1, y1).
y = 2 (x + 4)
Using the point (–4, 0), the equation of the line is y = 2(x + 4), or y = 2x + 8.
Lesson
Main
Lesson
11-6
Feature
Writing Rules for Linear Functions
LESSON 11-6
Course 3
Lesson Quiz
1. A telephone company charges $4.95 per month plus $.10 per
minute used. Write the rule to show how the monthly bill
depends on the minutes used.
y = 0.1x + 4.95
2. Write a function rule for the data in the table.
x
y
0
8
2
7
4
6
6
5
y = –0.5x + 8
Lesson
Main
Lesson
11-6
Feature
Writing Rules for Linear Functions
LESSON 11-6
Course 3
Lesson Quiz
3. An electrician charges $60 for a house call, plus $75 for each hour of
work. Write a function rule that shows how the total cost of the
electrician’s work y depends upon the number of hours x the electrician
works. What is the output for an input value of 4, and what does it
represent?
y = 75x + 60; $360; the electrician’s charge for 4 hours of work
Lesson
Main
Lesson
11-6
Feature
Quadratic and Other Nonlinear Functions
LESSON 11-7
Course 3
Problem of the Day
A blue cube is 3 times as high as a red cube. How many red cubes
would fit into the blue cube?
27
Lesson
Main
Lesson
11-7
Feature
Quadratic and Other Nonlinear Functions
LESSON 11-7
Course 3
Check Skills You’ll Need
(For help, go to Lesson 11–5.)
1. Vocabulary Review How do you know if a function is linear?
Graph each function.
2. y = 3x – 1
3. y = x + 7
4. y = – 2x – 5
5. y = – 4x + 2
Check Skills You’ll Need
Lesson
Main
Lesson
11-7
Feature
Quadratic and Other Nonlinear Functions
LESSON 11-7
Course 3
Check Skills You’ll Need
Solutions
1. The rate of change is constant.
3.
2.
4.
5.
Lesson
Main
Lesson
11-7
Feature
Quadratic and Other Nonlinear Functions
LESSON 11-7
Course 3
Additional Examples
Make a table and graph the quadratic function f(x) = x2 – 2x.
Use integers from –2 to 4 for inputs.
x
x2 – 2x
–2
(–2)2 – 2(–2)
–1
=
f(x)
= 4+4
=
8
(–1)2 – 2(–1)
= 1+2
=
3
0
(0)2 – 2(0)
= 0–0
=
0
1
(1)2 – 2(1)
= 1–2
=
–1
2
(2)2 – 2(2)
= 4–4
=
0
3
(3)2 – 2(3)
= 9–6
=
3
4
(4)2 – 2(4)
= 16 – 8
=
8
Quick Check
Lesson
Main
Lesson
11-7
Feature
Quadratic and Other Nonlinear Functions
LESSON 11-7
Course 3
Additional Examples
Quick Check
Suppose a driver is making a 120-mile trip. Make a table and
graph this situation using the relationship that 120 ÷ hours for trip =
speed for trip. Use 2, 3, 4, 5, and 6 hours for input.
Input
t
2
3
4
5
6
Output
y
120 ÷ 2 = 60
120 ÷ 3 = 40
120 ÷ 4 = 30
120 ÷ 5 = 24
120 ÷ 6 = 20
The data are continuous, so the curve is solid.
Lesson
Main
Lesson
11-7
Feature
Quadratic and Other Nonlinear Functions
LESSON 11-7
Course 3
Additional Examples
Write a quadratic function rule for the data in the table below.
Input x
–2
–1
0
1
2
Output y
1
–2
–3
–2
1
Input
x
(Input)2
x2
Output
y
–2
4
1
–1
1
–2
0
0
–3
1
1
–2
2
4
1
Lesson
Main
Compare each output
to (input)2.
Each output is less
than (input)2 by 3.
Lesson
11-7
Feature
Quadratic and Other Nonlinear Functions
LESSON 11-7
Course 3
Additional Examples
(continued)
Words output = (input) 2 – 3
Let x = input.
Let y = output.
Function
y
= x
2
–3
So, the function rule is y = x2 – 3.
Quick Check
Lesson
Main
Lesson
11-7
Feature
Quadratic and Other Nonlinear Functions
LESSON 11-7
Course 3
Lesson Quiz
For Exercises 1–2, use the function rule y = –x2 + 4x.
1. Make a table for the function.
x
y
0
0
1
3
2
4
3
3
4
0
2. Graph the function.
Lesson
Main
Lesson
11-7
Feature
Quadratic and Other Nonlinear Functions
LESSON 11-7
Course 3
Lesson Quiz
For Exercises 3–4, use the function rule y = x + 1 .
x
3. Make a table for the function.
x
y
1
2
2
2.5
3
3.3
4
4.25
5
5.2
4. Graph the function.
Lesson
Main
Lesson
11-7
Feature