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Algebra II - Exponential and Logarithmic Functions

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    Study Guide Algebra II – Exponential and Logarithmic Functions 1. Exponential Functions What Is an Exponential Function? An exponential function is a function where the variable is in the exponent . It has the general form: Where: • b is the base • b > 0 • b ≠ 1 • x is any real number Key Idea Exponential functions grow very quickly . For example: • (2 2 = 4) • (2 3 = 8) • (2 7 = 128) After only a few steps, the numbers become very large. That is why exponential growth is called rapid growth . Example 1 — Graphing (y = 2 x )

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    Study Guide Figure 1. Exponential growth of 2. To graph an exponential function, we first create a table of values . x y = (2 x ) Ordered Pair - 3 (1/8) ( - 3, 1/8) - 2 (1/4) ( - 2, 1/4) - 1 (1/2) ( - 1, 1/2) 0 1 (0,1) 1 2 (1,2) 2 4 (2,4) 3 8 (3,8) Steps to Graph 1. Plot these points on the coordinate plane. 2. Connect them smoothly. Important Observations • The graph passes through (0,1) .

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    Study Guide • The graph never touches the x - axis . • As (x) becomes more negative, (y) gets closer to 0 . The x - axis becomes a horizontal asymptote (the curve approaches it but never touches it). Example 2 — Graphing (f(x) = (1/2) x ) Figure 2. Exponential growth of 1/2. Now consider the function: Table of Values x (f(x)) Ordered Pair - 3 8 ( - 3,8) - 2 4 ( - 2,4) - 1 2 ( - 1,2) 0 1 (0,1) 1 1/2 (1,1/2)

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    Study Guide 2 1/4 (2,1/4) 3 1/8 (3,1/8) What Happens Here? Unlike (2 x ), this graph decreases as x increases . This is called exponential decay . Key Observations • The graph still passes through (0,1) . • The curve approaches the x - axis but never touches it . Important Property of Exponential Functions All exponential functions: contain the point: because Growth vs Decay Exponential Growth Occurs when: Example:

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    Study Guide The graph increases rapidly . Exponential Decay Occurs when: Example: The graph decreases as (x) increases. Example 3 — Graphing (y = 3 x ) and (x = 3 y ) Figure 3. In this graph, y = 3 x and x = 3 y . Now we compare two functions: • (y = 3 x ) • (x = 3 y ) Points for (y = 3 x )

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    Study Guide x y - 3 1/27 - 2 1/9 - 1 1/3 0 1 1 3 2 9 3 27 Ordered pairs: ( - 3, 1/27) ( - 2, 1/9) ( - 1, 1/3) (0, 1) (1, 3) (2, 9) (3, 27) Points for (x = 3 y ) Here we switch x and y values . y x - 3 1/27

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    Study Guide - 2 1/9 - 1 1/3 0 1 1 3 2 9 3 27 Ordered pairs: (1/27, −3) (1/9, −2) (1/3, −1) (1, 0) (3, 1) (9, 2) (27, 3) Relationship Between the Graphs The graphs of • (y = 3 x ) • (x = 3 y ) are inverse functions . Inverse graphs are mirror images across the line (y = x) . Key Characteristics of Exponential Functions

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    Study Guide 1. Domain: All real numbers 2. Range: Positive real numbers 3. The graph always passes through (0,1) 4. The x - axis is a horizontal asymptote 5. If b > 1 → growth 6. If 0 < b < 1 → decay Summary An exponential function has the form: where (b > 0) and (b ≠ 1). Key Ideas • Exponential functions grow or shrink very quickly . • Every exponential graph passes through (0,1) . • The x - axis acts as an asymptote . • If b > 1 , the graph shows growth . • If 0 < b < 1 , the graph shows decay . • Inverse exponential functions are reflections across (y=x) . 2. Quiz: Exponential Functions 1. Question For the function what is the value of y when (x = 4) ?

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    Study Guide Answer Choices • 12 • 27 • 81 Correct Answer : 81 Why This Is Correct Substitute (x = 4) into the function. So the value of y is 81 . 2. Question For the function what is the value of y when (x = - 5) ? Answer Choices • −243 • 1/81 • 1/243 Correct Answer

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    Study Guide Why This Is Correct Substitute (x = - 5). Use the negative exponent rule: So the value of y is 1/243 . 3. Question For the function which is its inverse function ? Answer Choices • (x = 3 y ) • (x = y 3 ) • (3 = x y ) Correct Answer

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