1. Exponential Functions and Graphs
Section 5.2
Exponential Functions and Graphs
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

2. Objectives Graph exponential equations and exponential functions.
Solve applied problems involving exponential functions and their graphs.

3. Exponential Function
The function f(x) = ax, where x is a real number, a > 0 and a  1, is called the exponential function, base a.
The base needs to be positive in order to avoid the complex numbers that would occur by taking even roots of negative numbers.
The following are examples of exponential functions:

4. Example
Graph the exponential function y = f (x) = 2x.

5. Example (continued)
As x increases, y increases without bound. As x decreases, y decreases getting close to 0; as x g ∞, y g 0.
The x-axis, or the line y = 0, is a horizontal asymptote. As the x-inputs decrease, the curve gets closer and closer to this line, but does not cross it.

6. Example Graph the exponential function Note
This tells us the graph is the reflection of the graph of y = 2x across the y-axis. Selected points are listed in the table.

7. Example (continued)
As x increases, the function values decrease, getting closer and closer to 0. The x-axis, y = 0, is the horizontal asymptote. As x decreases, the function values increase without bound.

8. Graphs of Exponential Functions
Observe the following graphs of exponential functions and look for patterns in them.

9. Example
Graph y = 2x – 2.
The graph is the graph of y = 2x shifted to right 2 units.

10. Example
Graph y = 5 – 0.5x .
The graph is a reflection of the graph of y = 2x across the y-axis, followed by a reflection across the x-axis and then a shift up 5 units.

11. Application
The amount of money A that a principal P will grow to after t years at interest rate r (in decimal form), compounded n times per year, is given by the formula

12. Example
Suppose that $100,000 is invested at 6.5% interest, compounded semiannually.
a. Find a function for the amount to which the investment grows after t years.
b. Graph the function.
c. Find the amount of money in the account at t = 0, 4, 8, and 10 yr.
d. When will the amount of money in the account reach $400,000?

13. Example (continued)
Since P = $100,000, r = 6.5%=0.65, and n = 2, we can substitute these values and write the following function

14. Example (continued)
b) Use the graphing calculator with viewing window [0, 30, 0, 500,000].

15. Example (continued)
We can compute function values using function notation on the home screen of a graphing calculator.

16. Example (continued)
We can also calculate the values directly on a graphing calculator by substituting in the expression for A(t):

17. Example (continued) Set 100,000(1.0325)2t = 400,000 and solve for t,
which we can do on the graphing calculator. Graph the equations
y1 = 100,000(1.0325)2t
y2 = 400,000
Then use the intersect method to estimate the first coordinate of the point of intersection.

18. Example (continued) Or graph y1 = 100,000(1.0325)2t – 400,000 and use
the Zero method to estimate the zero of the function coordinate of the point of intersection. Regardless of the method, it takes about years, or about 21 yr, 8 mo, and 2 days.

19. The Number e
e is a very special number in mathematics. Leonard Euler named this number e. The decimal representation of the number e does not terminate or repeat; it is an irrational number that is a constant; e  …

20. Example
Find each value of ex, to four decimal places, using the ex key on a calculator. a) e3 b) e0.23 c) e2 d) e1
a) e3 ≈
b) e0.23 ≈
c) e0 = 1
d) e1 ≈

21. Graphs of Exponential Functions, Base e Example
Graph f (x) = ex and g(x) = e–x.
Use the calculator and enter y1 = ex and y2 = e–x. Enter numbers for x.

22. Graphs of Exponential Functions, Base e - Example (continued)
The graph of g is a reflection of the graph of f across they-axis.

23. Example Graph f (x) = ex + 3.
The graph f (x) = ex + 3 is a translation of the graph of y = ex left 3 units.

24. Example Graph f (x) = e–0.5x.
The graph f (x) = e–0.5x is a horizontal stretching of the graph of y = ex followed by a reflection across the y-axis.

25. Example Graph f (x) = 1  e2x.
The graph f (x) = 1  e2x is a horizontal
shrinking of the graph of y = ex followed by a reflection across the y-axis and then across the x-axis, followed by a translation up 1 unit.