### Chapter 9 - Gordon State College

```9
Modeling
Our World
Chapter 9, Unit B, Slide 1
Unit 9B
Linear Modeling
Chapter 9, Unit B, Slide 2
Linear Functions
A linear function has a constant rate of change and a
straight-line graph.
 The rate of change is equal to the slope of the graph.
 The greater the rate of change, the steeper the slope.
 Calculate the rate of change by finding the slope
between any two points on the graph.
ra te o f ch a n g e = slo p e =
ch a n g e in d e p e n d e n t v a ria b le
ch a n g e in in d e p e n d e n t v a ria b le
Chapter 9, Unit B, Slide 3
Finding the Slope of a Line
To find the slope of a straight line, look at any two points and
divide the change in the dependent variable by the change in
the independent variable.
Chapter 9, Unit B, Slide 4
Example
A small store sells fresh pineapples. Based on data
for pineapple prices between \$2 and \$5, the
storeowners created a model in which a linear
function is used to describe how the demand
(number of pineapples sold per day) varies with the
price. For example, the point (\$2, 80 pineapples)
means that, at a price of \$2 per pineapple, 80
pineapples can be sold on an average day. What is
the rate of change for this function? Discuss the
validity of this model.
Chapter 9, Unit B, Slide 5
Example (cont)
Solution
Point 1 (\$2, 80)
Point 2 (\$5, 50)
Change in demand is
50 – 80 = –30
The change in demand is negative because
demand decreases from Point 1 to Point 2.
Chapter 9, Unit B, Slide 6
Example (cont)
rate of change =
change in dem and
change in price

 30
\$3

 10
\$1
For every dollar that the price increases, the number
of pineapples sold decreases by 10.
Chapter 9, Unit B, Slide 7
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
 rate of  

C hange in dependent variable  


change

  independent varaible 
Chapter 9, Unit B, Slide 8
Example
Using the linear demand function in Figure 9.12,
predict the change in demand for pineapples if the
price increases by \$3.
Solution
Independent variable = price of the pineapples
Dependent variable = demand for pineapples
Rate of change of demand with respect to price is
−10 pineapples per dollar.
change in demand = rate of change × change in price
= –10 × \$3
= –30 pineapples
Chapter 9, Unit B, Slide 9
Equations of Lines
General Equation for a Linear Function
dependent variable = initial value + (rate of change  independent variable)
Algebraic Equation of a Line
In algebra, x is commonly used for the independent
variable and y for the dependent variable. For a straight
line, the slope is usually denoted by m and the initial
value, or y-intercept, is denoted by b.
With these symbols, the equation for a linear function
becomes y = mx + b.
Chapter 9, Unit B, Slide 10
Slope and Intercept
For example, the
equation y = 4x – 4
represents a straight
line with a slope of 4
and a y-intercept of –4.
As shown to the right,
the y-intercept is where
the line crosses the
y-axis.
Chapter 9, Unit B, Slide 11
Varying the Slope
The figure to the right
shows the effects of
keeping the same
y-intercept but changing
the slope. A positive slope
(m > 0) means the line
rises to the right. A
negative slope (m < 0)
means the line falls to the
right. A zero slope (m = 0)
means a horizontal line.
Chapter 9, Unit B, Slide 12
Varying the Intercept
The figure to the right
shows the effects of
changing the y-intercept
for a set of lines that have
the same slope. All the
lines rise at the same
rate, but cross the y-axis
at different points.
Chapter 9, Unit B, Slide 13
Example: Rain Depth Equation
Use the function shown
to the right to write an
equation that describes
the rain depth at any
time after the storm
began. Use the
equation to find the
rain depth 4 hours after
the storm began.
Since m = 0.5 and b = 0, the function is y = 0.5x.
After 4 hours, the rain depth is (0.5)(4) = 2 inches.
Chapter 9, Unit B, Slide 14
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.
Step 2: Substitute the slope, m, and the numerical values of x
and y from either point into the equation y = mx + b and
solve for the y-intercept, b.
Step 3: Use the slope and the y-intercept to write the equation
in the form y = mx + b.
Chapter 9, Unit B, Slide 15
Example
Until about 1850, humans used so little crude oil
that we can call the amount zero—at least in
comparison to the amount used since that time. By
1960, humans had used a total (cumulative) of 600
billion cubic meters of oil. Create a linear model that
describes world oil use since 1850. Discuss the
validity of the model.
Solution
Data points: (1850, 0), (1960, 600)
Chapter 9, Unit B, Slide 16
Example (cont)
Data points: (1850, 0), (1960, 600)
Find the slope:
m 
600  0
1960  1850

600
 5 . 45 b illio n m
3
p e r ye a r
110
Looking for an equation of the form y = mt + b
m = 5.45
Find b.
0 = 5.45(1850) + b
b = –10,083 billion m3 per year
Equation: y = 5.45t – 10,083
Chapter 9, Unit B, Slide 17
Example (cont)
Validity as a model for oil consumption.
Draw the graph only for years after 1850.
Years prior to 1850, in which we assumed oil
consumption to be zero, are not in the domain of the
function.
The more significant issue is whether the rise in oil
consumption has been linear. The rate of oil
consumption tends to increase with population. In
fact, it has increased more rapidly than population
because the average person uses more oil now
than in the past.
Chapter 9, Unit B, Slide 18
Example (cont)
We know that population has risen exponentially
since the mid-19th century (see Unit 8C), so a better
model for oil consumption would be exponential
instead of linear. The blue curve in Figure 9.16
shows an exponential fit to the
two data points; we will
discuss the creation of
exponential models in the
next unit.