1. Growth and Decay Exponential Models

2. 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

3. 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:

4. 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:

5. 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

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

7. Exponential Growth vs. Decay GROWTHDECAY

8. 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.

9. 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

10. 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)?

11. 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.

12. 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