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Growth and Decay Exponential Models

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Differs from polynomial functions. Polynomial Functions have exponents of whole numbers Exponential Functions have “x” as the exponent. Exponential Model: f(x) = a∙b x, a > 0, b > 0, and x is any real number There are two kinds of exponential models Growth Decay

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Exponential Growth f(x) = a∙b x, where b > 1 Examples: f(x) = 5 x, g(x) = 2 ∙3 x Definition: as x increases so does f(x), and it increases very quickly. Graph:

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Exponential Decay f(x) = a∙b x, where 0 < b < 1 Examples: f(x) = (1/2) x, g(x) = 3 ∙(3/5) x Definition: as x increases, f(x) decreases quickly, but never reaches zero. Graph:

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Determine if the model is exponential growth or decay. 1. f(x) = 3 ∙2 x 2. g(x) = -5 ∙4 x 3. f(x) = 4 ∙(1/3) x 4. h(x) =.89 -x 5. g(x) = 7 -x 6. f(x) = (5/3) x

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Determine if the graph is exponential growth, decay, or neither. 1. 2. 3.

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Exponential Growth vs. Decay GROWTHDECAY

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Horizontal Asymptote Both, exponential growth & decay, have a horizontal asymptote. A horizontal asymptote is the line that f(x) approaches as x approaches positive of negative infinity. It also is the lower boundary of the range for the exponential functions.

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Identify the horizontal asymptote, domain, and range of the function. 1. f(x) = 2 x 2. h(x) = (2/3) x 3. f(x) = 4 (x+1) 4. f(x) = 5 x + 3

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Transformations Graph the following functions in your calculator. How to the graphs transform from f(x) to g(x)? f(x) = 2 x and g(x) = 2 x+1 How would f(x) transform to g(x)? f(x) = 2 x and g(x) = 2 x – 1 Summarize how you would describe the transformation of f(x) to g(x+a)?

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Transformations cont. Graph the following functions in your calculator. How to the graphs transform from f(x) to g(x)? f(x) = 3 x and g(x) = 3 x + 1 How would f(x) transform to g(x)? f(x) = 3 x and g(x) = 3 x – 1 Summarize how you would describe the transformation of f(x) to g(x) + a.

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Describe the transformation of f(x) to g(x). 1. f(x) = 4 x and g(x) = 4 x+5 2. f(x) = (1/3) x and g(x) = (1/3) x – 4 3. f(x) = 2∙8 x and g(x) = 2∙8 x-3 + 1 4. f(x) = 7 x+1 and g(x) = 7 x+3 – 6

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