1. Pre-AP Pre-Calculus Chapter 3, Section 1
Exponential and Logistic Functions

2. Function vs. Function What do you notice about the two functions?
𝑓 𝑥 = 𝑥 𝑓 𝑥 = 2 𝑥

3. Exponential Functions & Their Graphs
The functions 𝑓 𝑥 = 𝑥 2 and 𝑔 𝑥 = 2 𝑥 each involve a base raised to a power, but the roles are reversed.
For 𝑓 𝑥 = 𝑥 2 , the base is the __________, and the exponent is a ___________. This is a monomial and a power function.
For 𝑔 𝑥 = 2 𝑥 , the base is a _______, and the exponent is a _________. This is called an _________________________________.

4. Definition: Exponential Function
An exponential function can be written in the form: 𝑓 𝑥 =
where a & b are real number __________.
a is ____________ & the ________________.
b is ____________ and does not __________.
b is also called the _________.

5. Are they exponential functions?
𝑓 𝑥 = 3 𝑥
𝑔 𝑥 = 6𝑥 −4
ℎ 𝑥 =−2∙ 1.5 𝑥
𝑘 𝑥 =6∙ 5 𝜋

6. Compute the exponential function value
Use 𝑓 𝑥 = 2 𝑥
𝑓 4 =
𝑓 0 =
𝑓 −3 =
𝑓 =
𝑓 − 3 2 =

7. Finding an Exponential Function from its Table of Values
g(x)
h(x) -2
4/9
128 -1
4/3 32 4 8 1 12 2 36
1/2

8. Finding an Exponential Function from its Table of Values
f(x)
t(x) -2 6
108 -1 3 36
3/2 12 1
3/4 4 2
3/8
4/3

9. Graph of Exponential Functions
Graph each function in the viewing window [-2, 2] by [-1, 6]
𝑦 1 = 2 𝑥
𝑦 2 = 3 𝑥
𝑦 3 = 4 𝑥
𝑦 4 = 5 𝑥
Which point is common to all four graphs?
Analyze each graph for domain, range, extrema, continuity, increasing or decreasing, symmetry, asymptotes, and end behavior.

10. Graph of Exponential Functions
Graph each function in the viewing window [-2, 2] by [-1, 6]
𝑦 1 = 𝑥
𝑦 2 = 𝑥
𝑦 3 = 𝑥
𝑦 4 = 𝑥
Which point is common to all four graphs?
Analyze each graph for domain, range, extrema, continuity, increasing or decreasing, symmetry, asymptotes, and end behavior.

11. Exponential Growth & Decay

12. The Natural Base e 𝑓 𝑥 = 𝑒 𝑥 Domain: Range: Continuity: Increasing:
Symmetry:
Extrema:
Asymptotes:
End Behavior:

13. Facts about e The number e is _______________.
Since the graph is increasing, e is considered _____________________________.
Because of specific calculus properties, the function e is considered the _____________ _________ of exponential functions.
𝑓 𝑥 = 𝑒 𝑥 is considered the _____________________________________.

14. Logistic Functions & Their Graphs
Exponential growth is unrestricted. Meaning, _____________________________________.
In many growth situations, there is a limit to possible growth. A plant can only grow so tall. The number of goldfish in an aquarium is limited by the size of the aquarium.
In such growth situations, the beginning is exponential in manner, but slows down and eventually levels out.
These types of growth situations have horizontal asymptotes.

15. Logistic Growth Functions
Let a, b, c, and k be positive constants, with b < 1. A logistic growth function in x in a function that can be written in the form
𝑓 𝑥 = 𝑐 1+𝑎∙ 𝑏 𝑥
where the constant c is the ______________.
Unless otherwise noted, all logistic functions in the book are logistic growth functions.

16. Basic Function: The Logistic Function
𝑓 𝑥 = 1 1+ 𝑒 −𝑥
[-4.7, 4.7] by [-0.5, 1.5]
Domain:
Range:
Continuity:
Extrema:
Asymptotes:
End Behavior:

17. Graphing Logistic Growth Functions
Graph the function. Find the y-intercept and the horizontal asymptotes.
𝑓 𝑥 = 8 1+3∙ 0.7 𝑥

18. Graphing Logistic Growth Functions
Graph the function. Find the y-intercept and the horizontal asymptotes.
𝑓 𝑥 = 𝑒 −3𝑥

19. The Population of San Jose, California
Population Growth
Use the data in the table. Assuming the growth is exponential, when will the population of San Jose surpass 1 million persons?
The Population of San Jose, California
Year
Population
1990
782,248
2000
895,193

20. Modeling Dallas’s Population
Based on recent census data, a logistic model for the population of Dallas, t years after 1900, is as follows:
𝑃 𝑡 = 1,301, 𝑒 − 𝑡
According to this model, when was the population 1 million?

21. Ch 3.1 Homework
Page 286 – 288, #’s: 3, 4, 5, 8, 10, 13, 28, 42, 56, 61
10 total problems
Gray book: pg