1. Section 9B Linear Modeling
Pages

2. 9-B
Linear Functions
A linear function has a constant rate of change and a straight-line graph.
old example: The initial population of Straightown is 10, 000 and increases by 500 people per year.
Independent variable: year
Dependent variable: population
Population is a function of time(year).
The constant rate of change is 500 people per year.

3. old example: The initial population of Straightown is 10, 000 and increases by 500 people per year.
Graph
Data Table
Year
Straightown
10,000 5
12,500 10
15,000 15
17,500 20
20,000 40
30,000
Equation
P = 10, x t

4. We define rate of change of a linear function by:
where (x1,y1) and (x2,y2) are any two ordered pairs of the function.

5. Rate of change is ALWAYS 500 (people per year).
old example: The initial population of Straightown is 10, 000 and increases by 500 people per year.
Year
Straightown
10,000 5
12,500 10
15,000 15
17,500 20
20,000 40
30,000
= 500
= 500
= 500
= 500
Rate of change is ALWAYS 500 (people per year).

6. We define slope of a straight line by:
where (x1,y1) and (x2,y2) are any two points on the graph of the straight line.

9. Postage cost (dependent) 1 oz $0.37 2 oz $0.60 3 oz $0.83 4 oz $1.06
9-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.
Weight (independent)
Postage cost (dependent)
1 oz
$0.37
2 oz
$0.60
3 oz
$0.83
4 oz
$1.06
5 oz
$1.29
6 oz
$1.52
7 oz
$1.75
Rate of change is ALWAYS 0.23 (dollars per ounce).

10. Rate of change is $0.23 per oz.
9-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.
Weight
Postage cost
Difference
1 oz
$0.37
2 oz
$0.60
$0.23
3 oz
$0.83
4 oz
$1.06
5 oz
$1.29
6 oz
$1.52
7 oz
$1.75
Rate of change is $0.23 per oz.

11. 9-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.

12. 9-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.

13. 9-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.

14. 9-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.

15. For linear functions:
Slope = Rate of Change
Use any two ordered pairs (points on the graph) to calculate rate of change (slope).

16. How does rate of change (slope) affect steepness…
9-B
…the greater the rate of change (slope), the steeper the graph.

17. Independent variable: price Dependent variable: demand (of pineapples)
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
Independent variable: price
Dependent variable: demand (of pineapples)
Demand is a function of price.
($2,80) and ($5,50)

18. ($2, 80 pineapples) and ($5, 50 pineapples)
9-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.

19. ($2, 80 pineapples) and ($5, 50 pineapples).
9-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.

20. ($2, 80 pineapples) and ($5, 50 pineapples).
9-B
($2, 80 pineapples) and ($5, 50 pineapples).
For every dollar increase in price, the demand for pineapples decreases by 10.

21. For 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

22. The Rate of Change Rule (page546)
9-B
The Rate of Change Rule (page546)
change in dependent variable = (rate of change) x (change in independent variable)
ex3/545 Predict the change in demand for pineapples if the price increases by $3. change in demand = (-10 pineapples per dollar) x $ = pineapples
If the price of pineapples increases by $3, then the demand will decrease by 30 pineapples

23. More Practice
17/554 The water depth in a reservoir decreases at a rate of 0.25 inch per hour because of evaporation. How much does the water depth change in 6.5 hours?
19/554 A tree increases its diameter by 0.2 inches per year by adding annual rings. How much does the diameter of the tree increase in 4.5 years?

24. 9-B
General Equation for a Linear Function dependent = initial value + (rate of change x independent) or y = m x + b where m is slope and b is y intercept.

25. General Equation for a Linear Function
Straightown Population: m = 500 and initial value = 10000
P = t
[ Y = x ]
Pineapple Demand: m = -10 and initial value = 100
D = 100 – 10p
[ Y = 100 – 10x ]

26. More 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) At what snow depth will the snowplow be unable to move?
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?

27. More 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?

28. Linear Functions Constant Rate of Change Straight Line Graph
Dependent = Initial + Rate x Independent
Y = mX + b

29. 9-B
Homework:
Pages
# 12a-b, 14a-b, 18, 20, 24, 26, 28, 32