1. Chapter 2 – Linear and Exponential Functions
2.1 – Introducing Linear Models
2.2 – Introducing Exponential Models
2.3 – Linear Model Upgrades

2. 2.1
A linear function models any process that has a constant rate of change.
m =
The graph of a linear function is a straight line.
A linear function has the form:
y = f(x) = b + mx
where
f is the name of the function.
b is the starting value or y intercept (f(0)).
m is the constant rate of change or slope.
slope intercept form

3. 2.2
An exponential function models any process in which function values change by a fixed ratio or percentage.
An exponential function models any process in which the ratio of consecutive function values is constant.
The graph of an exponential function is curvy.
An exponential function has the form:
y = f(x) = c * ax
where
f is the name of the function.
c is the starting value or y intercept (f(0)).
a is the growth factor (ratio of consecutive function values)

4. Growth factor is 1.05 [100% + 5%]. f(t) = P*(1.05)t 2.2
A mutual fund grows by 5% each year. What is the growth factor?
t (years) 1 2
f ($) P
P *P
=(1.05)*P
(1.05)*P+.05*(1.05)*P = (1.05)*(1.05)*P
ratio of consecutive output values t
f(t+1)/f(t)
f(1) / f(0) = (1.05)*P / P = 1.05 1
f(2) / f(1) = (1.05)*(1.05)*P / (1.05)*P = 1.05
Growth factor is [100% + 5%].
f(t) = P*(1.05)t

5. 2.2
In 1992, U N Population Fund predicted “this decade will add a billion.” In 1992, the world population was 5.23 billion people and predicted to increase by million people that year.
linear model – assume constant rate of change of million people per year.
exponential model – assume constant percentage change [ 91.25/523 = 1.74%]
linear model W(t) = *t
where t is years since 1992 and W is in millions of people
exponential model W(t) = 5230*(1.0174)t
where t is years since 1992 and W is in millions of people

6. Exponential model almost matches prediction.

7. Depreciation of Jason’s $3000 sound system equipment
Method 1: Linear Model Depreciation by same amount($300) for lifetime of equipment(10 years).
B(t) = 3000 – 300*t
Method 2: Exponential Model
Double Declining Balance Depreciation – double the percentage from the linear method and apply each year.
300/3000 = 10%, so value decreases by 20% each year.
V(t) = 3000*(.80)t

8. At what point do both models predict the same value?

9. 34 = f(3) = 12 * a3 f(x) = c * ax 34 = 12 * a3 f(x) = 12 * ax
Exponential Models
What if percentage and growth factor are not explicitly stated?
What if only data points are given?
Example 29/84: A biology student counts 12 bacteria in a petri dish at the start of an experiment and 34 bacteria three hours later. Find the growth factor and write an exponential model for the number of bacteria t hours from the start of the experiment.
34 = f(3) = 12 * a3
f(x) = c * ax
34 = 12 * a3
f(x) = 12 * ax
34 / 12 = a3
f(x) = 12 * (1.42)x
2.83 = a3
(2.83)(1/3) = a
1.42 = a
growth factor is 1.42 [42% growth per hour]

10. Check Your Understanding 2.4 (page 70)
More Practice
Check Your Understanding 2.4 (page 70)
Check Your Understanding 2.5 (page 71)

11. More Practice

12. Section 2.3 Linear Model Upgrades
What if starting value is not explicitly stated?
What if rate of change is not explicitly stated?
Example: The average girl in the US is 102 cm tall at age 4. From then until age 13, her rate of growth is nearly constant at 6.1 cm per year
f(4) = 102 and m = 6.1.
f(x) = b + mx
b is not given
BUT 4 input gives 102 output
f(x) = *(x-4) for 4 ≤ x ≤ 13
adjust for “starting” values!

13. Section 2.3 Linear Model Upgrades
What if starting value is not explicitly stated?
What if rate of change is not explicitly stated?
Example 3/74: Write a formula for the linear function that contains the points (4,-10) and (-3,-12)
YIKES! no starting value and no slope.
Calculate slope: m = (-10 – (-12)) / (4 – (-3)) = 2/7
Adjust starting values: f(x) = (2/7) * (x – 4) or
f(x) = (2/7) * (x + 3)

14. The Point-Slope Form of a Line
Given any two points not in the same vertical line, write a linear function in two steps:
Calculate slope ( rise/run or Δy / Δx ). Call it m.
Use one of the points (x1 , y1 ) as “starting” and adjust using the template:
f(x) = y1 + m*( x – x1)
More Practice
Check your understanding 2.6 #2

15. Chapter 2 – Linear and Exponential Functions
HWp81: 1-6, 13-18, 21-23
HWp81: 7-12(class), 25-45
TURN IN: #13, #16, #22
TURN IN: #26(include Maple graph), #30, #41, #44(include Maple graph and solve command)