Objectives Compare properties of two functions. Estimate and compare rates of change.

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.

The rate of change for Gas Station A is about 3 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! Example 1 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.

Example 1 continued 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

Example 1 continued Mike’s average rate of change is m = 124 -20 = 104 = 26 5-1 4 John’s average rate of change is m = 204 -25 = 179 ≈ 25.57 8-1 7

Example 1 continued Remember to find the average rate of change over a data set, find the slope between the first and last data point. Helpful Hint

Example 3: Comparing Exponential and Polynomial Functions. Compare the end behavior of the functions f(x) = -x2 and g(x) = 4 logx.

Check It Out! Example 3 Compare the end behavior of the functions f (x) = 4x2 and g(x) = x3.

Example 2: Sketching Graphs of Functions Given Key Features. 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.

Example 2 continued 60 = 30a Add equations and solve. 2 = a 15 = 25(2) + 5b + 5 -8 = b h(t) = 2t2 – 8t + 5

minimum height: 3 ft below water level Example 2 continued Step 3 Find the maximum height of the function by finding the vertex. Graph the function and approximate the vertex. minimum height: 3 ft below water level

Remember, in the equation f(x) = a(x - h)2 + k, the point (h, k) represents the vertex. Helpful Hint

Lesson Quiz: Part I Compare the end behavior of each pair of functions. 1. f(x) = x and g(x) = -x4 f(x): as x approaches positive infinity, f(x) approaches positive infinity; as x approaches negative infinity, f(x) approaches negative infinity. g(x): as x approaches positive infinity, g(x) approaches negative infinity; as x approaches negative infinity, g(x) approaches negative infinity.

Lesson Quiz: Part 2 2. f(x) = 4ex and g(x) = log x f(x): as x → ∞, f(x) → ∞; as x→ –∞, f(x) → 0. g(x): as x →∞, g(x) → 1; as x → 0, g(x)→ –∞. 3. Find the equation of a quadratic function that describes the data in the table. f(x) = 3x2 -4x -10.