Essential Questions How do I use intervals of increase and decrease to understand average rates of change of quadratic functions?

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Essential Questions How do I use intervals of increase and decrease to understand average rates of change of quadratic functions?

Interpret Rates of Change of Quads 1.5 Interpret Rates of Change of Quads Example 1 Identify intervals of increase and decrease Graph the function y = x 2 + x – 2. Identify the intervals over which the graph increases and decreases. Solution You can see from the graph that as you move from left to right the value of the function __________ on the left side of the vertex and __________ on the right side of the vertex. decreases increases The x-coordinate of the vertex is The graph __________ over the interval x > _____ increases and __________ over the interval x < _____. decreases

Interpret Rates of Change of Quads Checkpoint. Graph the function. Identify the intervals over which the graph increases and decreases.

Interpret Rates of Change of Quads 1.5 Interpret Rates of Change of Quads Example 2 Calculate the average rate of change Calculate the average rate of change of the function y = 2x 2– 1 on the interval – 1 ≤ x ≤ 0. Solution Find the two points on the graph of the function that correspond to the endpoints of the interval. The average rate of change is the slope of the line that passes through these two points. The points are ________ and _________. The average rate of change is:

Interpret Rates of Change of Quads Example 3 Compare average rates of change Compare the average rates of change of y = 2x 2 and y = - x on 0 ≤ x ≤ 1. Solution The average rate of change of y = - x is the slope of the line, which is – 1. The points ________ and _________ correspond to the endpoints of the interval for y = 2x 2 . The average rate of change of y = 2x 2 on 0 ≤ x ≤ 1 is The average rate of change of the quadratic function is ____ times as great as the average rate of change of the linear function on the interval 0 ≤ x ≤ 1.

Interpret Rates of Change of Quads Checkpoint. Complete the following exercises.

Interpret Rates of Change of Quads Checkpoint. Complete the following exercises. On y = 2x, it is always 2. 2 is how many times as large as 3/2? The average rate of change of the line is 4/3 times as great as the average rate of change of the quadratic.

Interpret Rates of Change of Quads Example 4 Solve a real world problem Baseball The path of a baseball thrown at an angle of 40 can be modeled by y = -0.05x 2 +3.2x + 8 where x is the horizontal distance (in feet) from the release point and y is the corresponding height (in feet). Find the interval on which the height is increasing. Solution The height of the baseball will be increasing from the release point until it reaches its maximum height at the vertex. The x-coordinate of the vertex is So the height will be increasing on the interval ________________.

Interpret Rates of Change of Quads Checkpoint. For the quadratic model in Example 4, find the average rate of change on the given interval.