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

Example 1 Identify intervals of increase and decreaseIdentify intervals of increase and decreaseGraph the function Graph the function yy = = xx

22 + + xx – 2. Identify the – 2. Identify the intervals over which the graph increases and intervals over which the graph increases and decreases.decreases.

3.3 Interpret Rates of Change of Quads

SolutionYou 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.

decreasesincreases

The x-coordinate of the vertex is

1

1.

2 _______

a

bx

2 2

1

The graph __________ over the interval x > _____

and __________ over the interval x < _____.

increases 2

1

decreases2

1

,

2

1

2

1 ,

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

12

1 .1 2 xy

a

bx

2 2/12

0 0

102

1 2 y 1 V ,0 1 0 ,dec.

,0inc.

Example 2 Calculate the average rate of changeCalculate the average rate of changeCalculate the average rate of change of the Calculate the average rate of change of the function function yy = 2 = 2xx

22– 1 on the interval – 1 – 1 on the interval – 1 ≤ ≤ x x ≤ 0.≤ 0.

3.3 Interpret Rates of Change of Quads

SolutionFind 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.

1____2 2 y 11___ 2

___ 1

1___2 2 y 01___ 0

___ 1The points are ________ and _________. 1 ,1 1 ,0 The average rate of change is:

_____

r 1 10 1

21

2

Example 3 Compare average rates of changeCompare average rates of changeCompare the average rates of change of Compare the average rates of change of yy = 2 = 2xx

22 and and yy = = x x on 0 on 0 ≤ ≤ x x ≤ 1.≤ 1.

3.3 Interpret Rates of Change of Quads

SolutionThe 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 yy = 2 = 2xx

22 . The . The average rate of change of average rate of change of yy = 2 = 2xx

22 on 0 ≤ on 0 ≤ x x ≤ 1 is≤ 1 is

0 ,0

._________________________

r02

2 ,1

01 1

22

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 ≤ 0 ≤ x x ≤ 1.≤ 1.

2

3.3 Interpret Rates of Change of QuadsCheckpoint. Complete the following exercises.Checkpoint. Complete the following exercises.

of change of rate average theCalculate .2

2.21 interval on the 232 xxxy

2131 2 y 2 ,1

8 2232 2 y 8 ,228

r12 1

6 6

3.3 Interpret Rates of Change of QuadsCheckpoint. Complete the following exercises.Checkpoint. Complete the following exercises.

of change of rates average theCompare .3

2

.42on 2 and 14

1 2 xxyxy

1225.0 2 y 2 ,25 1425.0 2 y 5 ,4

25 r

24 2

3

On y = 2x, it is always 2.

2 is how many times as large as 3/2?

2 x2

3

3

23

23

4x

The average rate of change of the line is 4/3 times as great as the average rate of change of the quadratic.

Example 4 Solve a real world problemSolve a real world problemBaseballBaseball The path of a baseball thrown at an angle of 40 The path of a baseball thrown at an angle of 40 can be modeled by can be modeled by yy = = 0.050.05xx

22 +3.2 +3.2xx + 8 where + 8 where x x is the is the horizontal distance (in feet) from the release point and horizontal distance (in feet) from the release point and yy is is the corresponding height (in feet). Find the interval on the corresponding height (in feet). Find the interval on which the height is increasing.which the height is increasing.

3.3 Interpret Rates of Change of Quads

SolutionThe 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

_________________________________

x 05.02

2.3

32

So the height will be increasing on the interval ________________.320 x 23 ,0

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

320 .4 x8 802.3005.0 2 y 8 ,0

82.59 r

032 32

2.51

2.59 8322.33205.0 2 y .295 ,32

6.1

3.3 Interpret Rates of Change of Quads

Pg. 73, 3.3 #1-14

Textbook pg. 73 #2-12

even