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Exponential Growth Functions

Exponential Growth Functions. AII, 12.0: Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. . Recap of last week. What did we learn last week? Exponential Properties Revisited (7) 296

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Exponential Growth Functions

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  1. Exponential Growth Functions AII, 12.0: Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.

  2. Recap of last week • What did we learn last week? • Exponential Properties Revisited (7) 296 • Logarithm Properties (3) 442 • Special Logarithm properties (3 inverses)434 • Equality properties 448 • Change of Base Formula 440

  3. Overview for today Quiz (40 minutes MAX) Objectives • Logarithm expressions • Evaluate the expression • Properties of Logarithms • Expand expressions • Condense expressions • Solving Exponential and Logarithmic Equations • Equal powers property • Take the common logarithm • Extraneous solutions • Graph exponential growth functions

  4. General information you NEED Translating Exponential Functions • If |a|>1, the graph is vertically expanded • If 0<|a|<1, the graph is vertically compress • If a is negative, the graph is reflected about the x-axis • h represents an horizontal shift on the graph • k represents a vertical shift on the graph

  5. Objective: Exponential Growth Exponential Growth Graphs (pg412) • when a>0 and b>1 • The graph rises from left to right • The graph passes through (0,a) and (1,ab) • The domain is all real numbers • The range is y>0

  6. Example 1 1 1 SOLUTION 2 4 a 1 = y ab x = > b 1 x 0 1 2 3 2 1 y 1 2 4 8 – – Graph when and Graph the function y 2x. = Make a table of values for the function.

  7. Example 1 a 1 = y ab x = > b 1 Graph when and Plot the points from the table. Draw a curve that passes through the plotted points, as shown.

  8. Example 2 = • y 2 3x. = 2 2 1 3 9 4 SOLUTION a. Make a table of values. Then plot the points. > b 1 1 2 3 – – – x 0 1 2 y 2 6 27 Graph when and y ab x a 1 = a. Graph b. Graph • y 2x. = Draw a curve that passes through the plotted points, as shown at the right.

  9. Example 2 = 1 1 1 8 2 4 x 0 1 2 3 > b 1 y 1 2 1 – Graph when and y ab x a 1 = b. Make a table of values. Then plot the points. Draw a curve that passes through the plotted points, as shown at the right.

  10. Checkpoint y 5x = ANSWER y ab x = > b 1 2. ANSWER Graph when Graph the exponential function. 1. y 4x =

  11. Checkpoint • y 4 3x = ANSWER y ab x = > b 1 • y 0.1 2x = 4. ANSWER Graph when Graph the exponential function. 3.

  12. Properties of Exponential Functions State the Domain and Range Horizontal Asymptote • The Domain • Majority of the time all the real numbers • The range • Majority of the time from the horizontal asymptote and above • The horizontal asymptote is the minimum y-value. • Majority of the time it will be the k value from the general form:

  13. Example 3 y 2x = SOLUTION • Sketch the graph of , • which passes through • and Then translate the graph 3 units to the left. . ( ( ) ) 0, 1 2, 4 The graph passes through and ( ) ( ). – – 3, 1 1, 4 Graph an Exponential Function Graph the function. Describe the horizontal asymptote. State the domain and range. – a. b. y 2x + 3 y 2x 3 = = The graph’s asymptote is the x-axis.

  14. Example 3 • Again, sketch the graph of . • Then translate the graph 3 units • down. y 2x = > y 0. The graph passes through and ( ) – 0, 2 The graph’s asymptote is the line > y The domain is all real numbers, and the range is ( ). 2, 1 – y 3. = – 3. Graph an Exponential Function The domain is all real numbers, and the range is

  15. Checkpoint y 2; = > y 2 ANSWER domain: all real numbers, range: Graph an Exponential Function Graph the function. Describe the horizontal asymptote. State the domain and range. 5. y 3x 2 + =

  16. Checkpoint ANSWER > y domain: all real numbers, range: – y 5; = – 5 Graph an Exponential Function Graph the function. Describe the horizontal asymptote. State the domain and range. 6. – y 4x 5 =

  17. Checkpoint y 0; = > y 0 ANSWER domain: all real numbers, range: Graph an Exponential Function Graph the function. Describe the horizontal asymptote. State the domain and range. 7. y 3x 4 = –

  18. Checkpoint y 0; = > y 0 ANSWER y 2x + 2 = domain: all real numbers, range: Graph an Exponential Function Graph the function. Describe the horizontal asymptote. State the domain and range. 8.

  19. Conclusion Summary Assignment • What does the graph of an exponential growth function look like? • The graph of an exponential growth function of the form • Exponential Growth Pg415 #(17-39 ODD, 42-47EC) • Problems not finished will be left as homework.

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