Transcript Unit 9B - Gordon State College

Unit 9B
Linear Modeling
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,
change in dependent variable
rate of change  slope 
change in independen t variable
A GRAPH FOR THE SLOPE
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:
Change in


 dependent variable
Change in
  rate of  
  
  
  change   independen t variable



GENERAL EQUATION FOR A
LINEAR FUNCTION
 dependent   initial   rate of   independen t 

  
  
  

 variable   value   change   variable 
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.
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:
change in y
slope  m 
change in x
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.
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?
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?