1. Exponential Functions 4.3 **You might want graph paper**

2. Vocabulary An exponential function has the form y = ab x, where a ≠ 0 and the base b is a positive number other than 1. If a > 0 and b > 1, the function y = ab x is an exponential growth function – and b is called the growth factor or multiplier.

3. Evaluate Exp. Function Example 1.x = 1.44. x = 2.x = -35. x = 3.x = 1.686. x = 3.784

4. Graphing Exp. Functions Parent Plot the points ( 0, 1) and ( 1, base) The asymptote will be a horizontal line y = ? and it is always the shift up or down

5. Make a table of values.STEP 1 STEP 2 Plot the points from the table. Graph y = 2 x.

6. Graph the following on your graph paper.

7. Graph the function. SOLUTION Plot and (1, 2).Then, from left to right, draw a curve that begins just above the x -axis, passes through the two points, and moves up to the right. 0, 1 2 a.

8. Graph the function. b. y = – 5 2 x Plot (0, –1) and. Then,from left to right, draw a curve that begins just below the x -axis, passes through the two points,and moves down to the right. b.b. 1, – 5 2 SOLUTION

9. Graph SOLUTION Begin by sketching the graph y = 4 ● 2 x Then translate the graph: y = 4 ● 2 x – 1 – 3

10. Domain? Range? Asymptote?

11. Your Turn Graph each function below. State the Domain, Range and Asymptote. y = 5 x f(x) = 1/3 ● 4 x y = 2 x + 1 – 1

12. Your Turn: Answers Domain: all real #s Range: y > 0 Asym.: y = 0 Domain: all real #s Range: y > -1 Asym.: y = -1

13. Vocabulary An exponential decay function has the form y = ab x, where a > 0 and 0 < b < 1. The base b of an exponential decay function is called the decay factor.

14. Example 1 Graph y = State the domain and range. Give the equation for the asymptote.

15. Example 2 Graph y = -3(1/4) x. State the domain and range. Give the equation for the asymptote.

16. Example 3 Graph y = 2(2/3) x – 1 + 1. State the domain and range. Give the equation for the asymptote.

17. When b>1, f(x) = b x represents GROWTH When 0<b<1, f(x) = b x represents DECAY Growth vs. Decay

18. Use Functions Involving e The number denoted by the letter e is called the natural base e or the Euler number. –Think of e like you do π –e is approximately 2.718281828. A function of the form f(x) = ae rx is called a natural base exponential function.

19. Graph the function. Graph the parent first. State the domain and range. a) y = 4e -x + 5

20. Graph the function. State the domain and range. b) y = e 2(x+1) - 2

21. Exponential Equations Exponential equations are equations in which variable expressions occur as exponents. Property of Equality for Exponential Equations: If b is a positive number other than 1, then b x = b y if and only if x = y.

22. Examples