1. 8.7 Exponential and Power Function Models 8.8 Logistics Model

2. Writing an Exponential Function You can write an exponential function, if you have two points on the graph of the exponential graph. Procedure: 1. Use the two points to write two equations by substituting into the equation, y = ab x. 2.Take the first equation, solve for a. 3.Substitute the results of the first equation into the second equation. Solve for b. 4.Write the equation.

3. Example Write an exponential function of the form y = ab x whose graph passes through the points (1, 4) and (3, 16). Solution The equation is y = 2(2) x.

4. Finding an Exponential Model for a Set of Data x0123456789 y 14.713.512.912.411.911.410.910.410.09.6 The data appears to be an exponential decay. Procedure with the TI83 calculator. STAT EDIT - Put the x values in L1. Put the y values in L2. STAT CALC – select ExpReg - ENTER 2 nd LIST – selected L1, L2 - ENTER

5. Result y = a*b^x a = 14.29665308 b =.9558971055 So, the exponential model is y = 14.3(0.956) x.

6. Writing Power Function If we have two points on the graph, we can find the equation using the same procedure we used for exponential functions. Procedure: 1. Use the two points to write two equations by substituting into the equation, y = ax b. 2.Take the first equation, solve for a. 3.Substitute the results of the first equation into the second equation. Solve for b. 4.Write the equation.

7. Power Function Example A power function, y = ax b passes through the following points: (2, 16) and (3, 36). Find the equation of the power function

8. Logistic Growth Functions The problem with exponential and power growth functions that both models grow without bounds. Many growths in real life have a limit to the growth based on various constraints, such as available resources. The logistic model is one model that grows to a limit. This model is used frequently to model the spread of epidemics such as the Ebola virus or SARS.

9. The Graph of the Logistic Growth Function

10. Characteristics The form of the equation: The horizontal lines y = 0 and y = c are asymptotes. The y-intercepts is The domain is all real numbers, and the range is 1 < y <c. The graph increases from left to right. The point of maximum growth is To the left of the maximum growth the rate of growth is increasing. To the right of the maximum growth the rate of growth is decreasing.

11. Example You planted a seedling and kept track of its height h (in centimeters) over time t (in weeks). Use the data to find a model that gives h as a function of t. t012345678 h5122639518894103112 Procedure with the TI83 calculator. STAT EDIT - Put the x values in L1. Put the y values in L2. STAT CALC – select Logistic - ENTER 2 nd LIST – selected L1, L2 - ENTER

12. Result y = c/(1 + ae^(-bx)) a = 18.18103216 b =.7317420617 c = 116.8290157

13. Graph of the Example