1. 4.2 Exponential Functions and Equations
Simplify expressions and equations involving rational exponents.
Explore Exponential functions and their graphs.

2. A function that can be expressed in the form
and is positive, is called an Exponential Function.
Exponential Functions with positive values of x are increasing, one-to-one functions.
The parent form of the graph has a y-intercept at (0,1) and passes through (1,b).
The value of b determines the steepness of the curve.
The function is neither even nor odd. There is no symmetry.
There is no local extrema.

3. More Characteristics of
The domain is
The range is
End Behavior: As
The y-intercept is
The horizontal asymptote is
There is no x-intercept.
There are no vertical asymptotes.
This is a continuous function.
It is concave up.

4. How would you graph How would you graph Domain: Range: Y-intercept:
Horizontal
Asymptote:
Inc/dec?
Concavity?
How would you graph
Domain:
Range:
Y-intercept:
Horizontal
Asymptote:
Inc/dec?
Concavity?

5. Recall that if then the graph of is a reflection of about the y-axis.
Thus, if then
Domain:
Range:
Y-intercept:
Horizontal
Asymptote:
Inc/dec?
Concavity?

6. How does b affect the function?
If b > 1, then
f is an increasing function,
and
If 0 < b < 1, then
f is a decreasing function,
and

7. How would you graph
Is this graph increasing
or decreasing?
Notice that the reflection is decreasing, so the end behavior is:

8. How does b affect the function?
If b>1, then
f is an increasing function,
and
If 0<b<1, then
f is a decreasing function,
and

9. Transformations
Exponential graphs, like other functions we have studied, can be dilated,
reflected and translated.
It is important to maintain the same base as you analyze the transformations.
x-axis
Vertical stretch 3
Vertical shift down 1
Vertical shift up 3

10. More Transformations Reflect about the Vertical shrink
Horizontal shift
Horizontal shift
Vertical shift
Vertical shift
Domain:
Domain:
Range:
Range:
Horizontal
Asymptote:
Horizontal
Asymptote:
Y-intercept:
Y-intercept:
Inc/dec?
Inc/dec?
Concavity?
Concavity?

11. The number e
The letter e is the initial of the last name of Leonhard Euler ( )
who introduced the notation.
Since has special calculus properties that simplify many
calculations, it is the natural base of exponential functions.
The value of e is defined as the number that the expression
approaches as n approaches infinity.
The value of e to 16 decimal places is
The function is called the Natural Exponential Function

12. Domain:
Range:
Y-intercept:
H.A.:
Continuous
Increasing
No vertical asymptotes
and

13. Transformations Vertical stretch Reflect @ x-axis.
Horizontal shift left 2.
Vertical shift
Vertical shift down 1.
Vertical shift up 2
Domain:
Range:
Y-intercept:
H.A.:
Domain:
Range:
Y-intercept:
H.A.:
Domain:
Range:
Y-intercept:
H.A.:
Inc/dec?
Inc/dec?
Inc/dec?
Concavity?
Concavity?
Concavity?

14. Exponential Equations: The Rules of Exponents

15. Exponential Equations
Equations that contain one or more exponential expressions are called exponential equations.
Steps to solving some exponential equations:
Express both sides in terms of same base.
When bases are the same, exponents are equal.
i.e.:

16. Exponential Equations
Sometimes it may be helpful to factor the equation to solve:

17. Exponential Equations
Try:
1) )

18. Interest Problems 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
If you deposit $10,000 in an account that pays 4% interest compounded annually, how much money will you have in your account at the end of 15 years? Write an exponential function that represents this situation.
𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡

19. Interest Problems – part 2
If you deposit $10,000 in an account that pays 4% interest compounded quarterly, how much money will you have in your account at the end of 15 years? Write an exponential function that represents this situation.
𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡

20. Interest Problems – part 3
If you deposit $10,000 in an account that pays 4% interest compounded continuously, how much money will you have in your account at the end of 15 years? Write an exponential function that represents this situation.
𝐴=𝑃 𝑒 𝑟𝑡

21. Exponential Growth and Decay Problems
In 1995, there were 85 rabbits in Central Park.  The population increased by 12% each year.  How many rabbits were in Central Park in 2005?

22. In 2000, 50 grams of radium were stored
In 2000, 50 grams of radium were stored.  The half-life of radium is 1,620 years.  How many grams of radium remains after 4860 years?  Remember, half-life is the amount of time it takes for half of the amount of a substance to decay.