Chapter6 6-2 comparing functions
Objectives • Compare properties of two functions. • Estimate and compare rates of change.
Comparing functions • The graph of the exponential function y=0.2491e0.0081x approximates the population growth in Baltimore, Maryland.
Comparing functions • The graph of the exponential function y=0.0023e0.0089x approximates the population growth in Hagerstown, Maryland. The trends can be used to predict what the population will be in the future in each city. In this lesson you will compare the graphs of linear, quadratic and exponential functions.
Example 1: Comparing the Average Rates of Change of Two Functions. • George tracked the cost of gas from two separate gas stations. The table shows the cost of gas for one of the stations and the graph shows the cost of gas for the second station. Compare the average rates and explain what the difference in rate of change represents.
solution The rate of change for Gas Station A is about 3.0. The rate of change for Gas Station B is about 2.9. The rate of change is the cost per gallon for each of the Stations. The cost is less at Gas Station B.
Check it out !! • John and Mike opened savings accounts on the same day. They did not deposit any money initially, but deposited each week as shown by the graph and the table. Compare the average rates of change and explain what the rates represent in this situation.
Mike’s average rate of change is 26. John’s average rate of change is ≈ 25.57. The rate of change is the average amount of money saved per week. In this case, Mike’s rate of change is larger than John’s, so he saves about $0.43 more than John per week Mike’s average rate of change ism = 124 -20 = 104 = 26 5-1 4 John’s average rate of change ism = 204 -25 = 179 ≈25.57 8-1 7
Example#3 • The graph for the height of a diving bird above the water level, h(t), in feet after t seconds passes through the points (0, 5), (3, -1), and (5,15). Sketch a graph of the quadratic function that models the situation. Find the point that represents the minimum height of the bird.
solution • Step 1 Use the points to find the values of a, b, and c in the function h(t) = at2 + bt + c. • (t, h(t))h(t) = at2 + bt + cSystem in a, b, c • (0,5) 5 = a(0)2 + b(0) + c • (3, -1) - 1 = a(3)2 + b(3) + c • (5, 15) 15 = a(5)2 + b(5) + c 5 = c -1 = 9a+3b+c 15=25a+5b+c
Solution minimum height: 3 ft below water level
Check it out!!! • The height of a model rocket after launch is tracked in the table. Find and graph a quadratic function that describes the data. The maximum height is approximately 61 feet.
Videos • Lets watch some videos
Student guided practice • Lets do problems 2-6 in your book page 418