1. Unit 9B
Linear Modeling

2. LINEAR FUNCTIONS
A linear function has a constant rate of change and a straight-line graph. For all linear functions,
The rate of change is equal to the slope of the graph.
The greater the rate of change, the steeper the slope.
We can calculate the rate of change by finding the slope between any two points on the graph. That is,

3. A GRAPH FOR THE SLOPE

4. THE RATE OF CHANGE RULE
The rate of change rule allows us to calculate the change in the dependent variable from the change in the independent variable:

5. GENERAL EQUATION FOR A LINEAR FUNCTION

6. ALGEBRAIC EQUATION FOR A LINE
The algebraic equation of a line is:
y = mx + b,
where m is the slope, and b is the initial value or y-intercept.

7. CREATING A LINEAR FUNCTION FROM TWO DATA POINTS
Step 1: Let x be the independent variable and y be the dependent variable. Find the change in each variable between the two given points and use these changes to calculate the slope, or rate of change:
Step 2: Substitute this slope and the numerical values of x and y from either data point into the equation y = mx + b. You can then solve for the y-intercept b.
Step 3: Now use the slope and y-intercept to write the equation of the linear function in the form y = mx + b.

8. EXAMPLE
A cereal company finds that the number of people who will buy one of its products in the first month that it is introduced is related to the amount of money spent on advertisings. If it spends $40,000 on advertising , then 100,000 boxes of cereal will be sold, and if it spends $60,000, then 200,000 boxes will be sold.
(a) Find a linear model describing this relationship. Interpret the slope as a rate of change.
(b) If $95,000 is spent on advertising, how many boxes of cereal will be sold?
(c) How much advertising in needed to sell 325,000 boxes of cereal?
(d) Is a linear model a good one for this relationship?

9. EXAMPLE
An insurance company has actuarial data which shows that person who is 15 years old has 62.3 years remaining lifetime and that a person who is 65 years old has 17.7 years remaining lifetime.
(a) Find a linear model describing this relationship. Interpret the slope as a rate of change.
(b) Based on your model, what is the remaining lifetime of a person who is 25 years old?
(c) If a person has 20 years remaining lifetime, how old is the person?
(d) Is a linear model a good one for this relationship?