Title: Section 9B Linear Modeling
1Section 9BLinear Modeling
2Linear Functions
9-B
A linear function describes a relation
betweenindependent (input) and dependent
(output) variables with a constant rate of
change (and a straight-line graph).
ExamplesStraightown population as a function of
time. Postage cost as a function of
weight. Pineapple demand as a function of price.
3We define rate of change of a linear function by
where (x1,y1) and (x2,y2) are any two ordered
pairs of the function.
4We define slope of a straight line by
where (x1,y1) and (x2,y2) are any two points on
the graph of the straight line.
59-B
General Equation for a Linear Functiondependent
initial value (rate of change x
independent)ory m x b where m is
slope and b is y intercept.
6old example The initial population of
Straightown is 10, 000 and increases by 500
people per year.
Graph
Data Table
7old example The initial population of
Straightown is 10, 000 and increases by 500
people per year.
500
500
500
500
Rate of change is ALWAYS 500 (people per year).
Initial population is 10000 (people).
Linear Function P 10000 500xt
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10Slope of the straight line graph is
500. Y-intercept of the straight line graph is
10000. Linear Function P 10000 500xt
11Linear Function P 10000 500xt What is the
population after 25 years? When will the
population be 28,000?
12Example The table below gives the cost of US
mail based on weight. What is the rate of change?
Graph the cost as a function of weight and
determine the slope.
9-B
Rate of change is ALWAYS 0.23 (dollars per ounce).
Initial cost is ??? (dollars).
Linear Function C ?? .23xw
139-B
Example The table below gives the cost of US
mail based on weight. What is the rate of change?
Graph the cost as a function of weight and
determine the slope.
149-B
Example The table below gives the cost of US
mail based on weight. What is the rate of change?
Graph the cost as a function of weight and
determine the slope.
159-B
Example The table below gives the cost of US
mail based on weight. What is the rate of change?
Graph the cost as a function of weight and
determine the slope.
169-B
Example The table below gives the cost of US
mail based on weight. What is the rate of change?
Graph the cost as a function of weight and
determine the slope.
17Slope of the straight line graph is
0.23. Y-intercept of the straight line graph is
.37-.23 .14. Linear Function C 0.14 0.23xw
18Linear Function C 0.14 0.23xw How much will
it cost to mail 6.5 ounces? How many ounces can
be mailed for 1.10?
19For linear functions Slope Rate of
Change Use any two ordered pairs (points on the
graph) to calculate rate of change (slope).
20How does rate of change (slope) affect steepness
9-B
the greater the rate of change (slope), the
steeper the graph.
219-B
ex2/545 A linear function is used to describe
how the demand for pineapples varies with the
price. We know at a price of 2, the demand is
80 pineapples and at a price of 5, the demand is
50 pineapples. Find the rate of change (slope)
for this function and then graph the function.
Express as a linear function.
Independent variable price Dependent variable
demand (of pineapples) Demand is a function of
price. (2,80) and (5,50)
229-B
ex2/545 A linear function is used to describe
how the demand for pineapples varies with the
price. We know at a price of 2, the demand is
80 pineapples and at a price of 5, the demand is
50 pineapples. Find the rate of change (slope)
for this function and then graph the function.
Express as a linear function.
(2, 80 pineapples) and (5, 50 pineapples)
For every dollar increase in price, the demand
for pineapples decreases by 10.
239-B
ex2/545 A linear function is used to describe
how the demand for pineapples varies with the
price. We know at a price of 2, the demand is
80 pineapples and at a price of 5, the demand is
50 pineapples. Find the rate of change (slope)
for this function and then graph the function
(2, 80 pineapples) and (5, 50 pineapples).
For every dollar increase in price, the demand
for pineapples decreases by 10.
249-B
(2, 80 pineapples) and (5, 50 pineapples).
For every dollar increase in price, the demand
for pineapples decreases by 10.
Slope of the straight line graph is
-10. Y-intercept of the straight line graph is
100. Linear Function D 100 -10xp
259-B
Linear Function D 100 -10xp What is the
demand for pineapples if the price is 8.50? If
the demand is 75, what is the corresponding price?
26For linear functions Slope Rate of Change Use
any two ordered pairs (points on the graph) to
calculate rate of change (slope). Postive Slope
Negative Slope
27More Practice
23/555 The price of a particular model car is
12,000 today and rises with time at a constant
rate of 1200 per year. A) Clearly identify
independent and dependent variable. B) Find a
linear equation to describe the situation. C)
How much will a new car cost in 2.5 years.
25/555 A snowplow has a maximum speed of 30
miles per hour on a dry highway. Its maximum
speed decreases by 0.5 miles per hour for every
inch of snow on the highway. A) Clearly identify
independent and dependent variable. B) Find a
linear equation to describe the situation. C)
How much will a new car cost in 2.5 years?
27/555 You can rent time on computers at the
local copy center for 5 setup charge and an
additional 3 for every 5 minutes. A) Clearly
identify independent and dependent variable. B)
Find a linear equation to describe the
situation. C) How much time can you rent for
15?
28More Practice
29/555 Suppose that you were 20 inches long at
birth and 4 feet tall on your tenth birthday.
A) Clearly identify independent and dependent
variable. B) Find a linear equation to describe
the situation. C) Use the equation to predict
your height at ages 2,6,20,50. D) Comment on
the validity of the model.
31/555 A YMCA fundraiser offers raffle tickets
for 5 each. The prize for the raffle is a 350
television set, which must be purchased with
proceeds from the ticket sales. Find an equation
that gives the profit/loss for the raffle as it
varies with the number of tickets sold. How many
tickets must be sold for the raffle sales to
equal the cost of the prize?
33/555 A 1000 washing machine is depreciated for
tax purposes at a rate of 50 per year. Find an
equation for the depreciated value of the washing
machine as it varies with time. When does the
depreciated value reach 0?
29Linear Functions
- Dependent Initial Rate x Independent
- Y mX b
309-B
- Homework
- Pages 553-555
- 12a-b, 14a-b, 24, 26, 28, 32