Transcript 2.4 ppt - Linear functions and models

Warm Up
1. Find the average rate of change of h(x) = x² - 2x + 3
a. From -1 to 1
b. From 0 to 2
c. From 2 to 5
*Hint: avg rate of change = Slope = [f(b) – f(a)] / [b – a]
a is the first number and b is the second number (read
from left to right).
2. Graph on your calculator and indicate intervals of
increase/decrease for f(x) = x³ - 3x² + 5 (-1,3)
2.4 Using Linear Models
1. Modeling Real-World Data
2. Predicting with Linear Models
(Find Lines of Best Fit – hand
drawn and w/graphing calc).
3. Graph linear equations.
1) Modeling Real-World Data
Big idea…
Use linear equations to create graphs of
real-world situations. Then use these
graphs and equations to make predictions
about past and future trends.
1) Modeling Real-World Data
Example 1:
There were 174 words typed in 3
minutes. There were 348 words typed
in 6 minutes. How many words were
typed in 5 minutes?
1) Modeling Real-World Data
x = independent
y = dependent
(x, y) = (time, words typed )
400
300
(x1, y1) = (3, 174)
200
(x2, y2) = (6, 348)
100
(x3, y3) = (5, ?)
1
2 3 4 5 6
Time (minutes)
Solution: y = 58x, y = 58(5)
total words = 290
2) Predicting with Linear Models
• You can extrapolate with linear models to
make predictions based on trends.
• Extrapolate - extend (a graph, curve, or
range of values) by determining unknown
values from trends in the known data.
2) Predicting with Linear Models
Example 1:
After 5 months the number of subscribers
to a newspaper was 5730. After 7 months
the number of subscribers was 6022.
Write an equation for the function. How
many subscribers will there be after 10
months?
2) Predicting with Linear Models
(x, y) = (months, subscribers)
(x1, y1) = (5, 5730)
8000
(x2, y2) = (7, 6022)
6000
(x3, y3) = (10, ?)
4000
2000
Equation:
2
4
6
Time
(months)
8
10
y = mx + b
2) Predicting with Linear Models
(x, y) = (months, subscribers)
(x1, y1) = (5, 5730)
8000
(x2, y2) = (7, 6022)
6000
(x3, y3) = (10, ?)
4000
2000
Equation:
2
4
6
Time
(months)
8
10
y = mx + b
2) Predicting with Linear Models
(x, y) = (months, subscribers)
(x1, y1) = (5, 5730)
8000
(x2, y2) = (7, 6022)
6000
(x3, y3) = (10, ?)
4000
2000
Equation:
2
4
6
Time
(months)
8
10
y = mx + b
2) Predicting with Linear Models
(x, y) = (months, subscribers)
(x1, y1) = (5, 5730)
8000
(x2, y2) = (7, 6022)
6000
(x3, y3) = (10, 6460)
4000
y-intercept
Equation: y = mx + b
2000
2
4
6
Time
(months)
8
10
y = 146x + 5000
y = 146(10) + 5000 = 6460
Total subscribers = 6460
Scatter Plots
• Connect the dots with a trend line (line of
best fit) to see if there is a trend in the
data.
Types of Scatter Plots
Strong, positive correlation
Weak, positive correlation
Types of Scatter Plots
Strong, negative correlation
Weak, negative correlation
Types of Scatter Plots
No correlation
Scatter Plots
Example 1:
The data table below shows the relationship between hours
spent studying and student grade.
a)
Draw a scatter plot. Decide whether a linear model is
reasonable.
b)
Draw a line of best fit. Write the equation for the line.
Hours
studying
3
1
5
4
1
6
Grade
(%)
65
35
90
74
45
87
Scatter Plots
(x, y) = (hours studying, grade)
(3, 65)
100
(1, 35)
90
(5, 90)
80
70
(4, 74)
60
(1, 45)
50
(6, 87)
40
30
Equation:
1
2 3 4 5
Hours studying
6
y = mx + b
Scatter Plots
(x, y) = (hours studying, grade)
(3, 65)
100
(1, 35)
90
(5, 90)
80
70
(4, 74)
60
(1, 45)
50
(6, 87)
40
30
1
2 3 4 5
Hours studying
6
a) Based on the graph, is a linear
model reasonable?
Scatter Plots
(x, y) = (hours studying, grade)
(3, 65)
100
(1, 35)
90
(5, 90)
80
70
(4, 74)
60
(1, 45)
50
(6, 87)
40
30
b) Equation: y = mx + b
1
2 3 4 5
Hours studying
6
y = 10.45x + 31.16
Phone Charges
The monthly cost C, in dollars, of a certain cellular phone
plan is given by the function C(x) = 0.38x + 5, where x is
the number of minutes used on the phone.
a.) What is the cost if you talk on the phone for 50 minutes?
b.) Suppose your monthly bill is $29.32. How many
minutes did you use the phone?
C.) Suppose that you budget yourself $60 per month for the
phone. What is the maximum number of minutes that you
can talk?
Phone Charges
The monthly cost C, in dollars, of a certain cellular phone
plan is given by the function C(x) = 0.38x + 5, where x is
the number of minutes used on the phone.
a.) What is the cost if you talk on the phone for 50 minutes?
C(50) = 0.38 (50) + 5 = $24.00
b.) Suppose your monthly bill is $29.32. How many
minutes did you use the phone? 29.32 = 0.38x + 5
x = 64 minutes
C.) Suppose that you budget yourself $60 per month for the
phone. What is the maximum number of minutes that you
can talk? 60 ≥ 0.38x + 6
144.73 ≥ x; x ≤ 144 mins
Supply and Demand
Suppose the quantity supplied, S, and quantity demanded, D,
of cellular telephones each month are given by the following
functions, where p is the price(in dollars) of the telephone:
S(p) = 60p – 900
and
D(p) = -15p + 2850
The equilibrium price is the price at which S(p) = D(p)
a.) Find the equilibrium price. What is the equilibrium quantity,
the amount demanded (or supplied), at the equilibrium price?
b.) Determine the prices for which quantity supplied is greater
than quantity demanded. That is, solve S(p) > D(p)
c.) Graph S(p) and D(p) and label the equilibrium price.
Supply and Demand
Suppose the quantity supplied, S, and quantity demanded, D,
of cellular telephones each month are given by the following
functions, where p is the price(in dollars) of the telephone:
S(p) = 60p – 900
and
D(p) = -15p + 2850
The equilibrium price is the price at which S(p) = D(p)
a.) Find the equilibrium price. What is the equilibrium quantity,
the amount demanded (or supplied), at the equilibrium price?
equilibrium price = $50, S(50) = D(50) = 2100 phones
b.) Determine the prices for which quantity supplied is greater
than quantity demanded. That is, solve S(p) > D(p)
60p – 900 > -15p + 2850
p > $50
c.) Graph S(p) and D(p) and label the equilibrium price.
c.) Graph S(p) and D(p) and label the equilibrium price.
Assignment
2.4 #1
p.101 1-10, 11-35 by 4’s