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Exponential Functions

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

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Exponential Functions

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

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

  3. Evaluate Exp. Function • Example • x = 1.4 4. x = • x = -3 5. x = • x = 1.68 6. 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. Graphy = 2x. STEP 1 Make a table of values. STEP 2 Plot the points from the table.

  6. Graph the following on your graph paper.

  7. 1 a. 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. 2 0, Graph the function. SOLUTION

  8. 1,– x b. y = – 5 5 b. 2 2 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. Graph the function. SOLUTION

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

  10. Domain? • Range? • Asymptote?

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

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

  13. Vocabulary An exponential decay function has the form y = abx, 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. Growth vs. Decay When b>1, f(x) = bx represents GROWTH When 0<b<1, f(x) = bx represents 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 • eis approximately 2.718281828. • A function of the form f(x) = aerx 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 = e2(x+1) - 2

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

  22. Examples

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