Chapter 9B SBM Modeling Our World

1
Chapter 9B SBMModeling Our World

• Linear Modeling

2
What is Linear Modeling?

• In 9A we represented functions with tables and
  graphs.
• In lesson 9B, we will turn our focus to
  representing functions with equations.
• These basic principles will be those of
  straight-line graphs called linear functions.

3
Linear Functions

• Whenever a graph is a straight line and is
  dealing with an independent and dependent
  variable, it is a linear function.

4
Slope (m)

• The slope of a line is defined as the amount that
  a graph rises vertically for a given distance
  that it runs horizontally. Or in other words
  rise over run.
• Take the change in y and divide it by the change
  in x.
• y-intercept form y mx b
• m slope and b y-intercept

5
Slope of Lines
The slope of a line through (x1 , y1) and (x2 ,
y2) with x1 not equal to x2, is y2 y1 x2
x1.
Ex. Find the slope (m) of a line through (7,5)
and (2,4).
Answer 1/5
6
The Change in the Dependent Variable

• The Change in the Dependent Variable is
  calculated by
• See example 3 on page 514.

(rate of change) X (change in indep. variable)
7
Equations for a Linear Function

• To generalize a linear function, you need
• A dependent variable
• An independent variable
• The initial value of the dependent variable when
  time (t) 0.
• The rate of change of N with respect to time or
  the change in N.

8
General Equation for a Linear Function

• The General Equation for a Linear Function is
  calculated by
• See examples 4 - 5 on pages 517-18.

Dependent
Variable Initial value
(rate of change) X (indep. variable)
9
Linear Functions from 2 Data Points

• Suppose we have 2 data points and want to find a
  linear function that fits them.
• You find the equation for the linear function by
  using the 2 data points to determine the rate of
  change (slope) and the initial value of the
  function.
• The following 3 steps summarize the process.

10
Creating a Linear Function from 2 Data Points

• Step 1 Let x be the independent variable and y
  be the dependent variable. Find the change in
  each variable between the 2 given points, and use
  these changes to calculate the slope (or rate of
  change).
• Slope

change in y
change in x
11
Steps (cont)

• Step 2 Substitute the slope (from step 1) and
  the numerical values of x and y from either data
  point into the y-intercept equation.
• You can then solve for the y-intercept, and slope
  because it will be the only unknown in the
  equation.
• Step 3 Now use the slope and y-intercept to
  write the equation of the linear function in the
  form of y mx b.
• See example 7 on pages 519-20.

12
Homework

• 9B 2 4, 8 12, 14 18, 20 22.
• Extra Credit 5 points possible for each
• 9B s 28, 30, 32
• (On separate sheet of paper please)