1. Objective: Use rate of change to solve problems. Find the slope of a line.

2.  Rate of change is a ratio that describes, on average, how much one quantity changes with respect to a change in another quantity.  If x is the independent variable and y is the dependent variable, then

3.  Use the table to find the rate of change. Then explain its meaning.  The car is travelling at the rate of 38 miles per hour. Time Driving (h) Distance Traveled (mi) XY 276 4152 6228

4.  Choose the best answer for the following. ◦ The table shows how the cost changes with the number of minutes used. Use the table to find the rate of change. Explain the meaning of the rate of change. A.Rate of change is 0.05 / 1. This means that it costs $0.05 per minute to use the cell phone. B.Rate of change is 5 / 1. This means that it costs $5 per minute to use the cell phone. C.Rate of change is 0.5 / 1. This means that it costs $0.50 per minute to use the cell phone. D.Rate of change is 0.20 / 1. This means that it costs $0.20 per minute to use the cell phone. Minutes Used Cost ($) XY 201 402 603

5.  So far, you have seen rates of change that are constant.  Many real-world situations involve rates of change that are not constant.

6.  The graph below shows the number of U.S. passports issued in 2002, 2004, and 2006. a.Find the rates of change for 2002-2004 and 2004-2006. b.Explain the meaning of the rate of change in each case. 2002-2004: There was an increase of 0.95 million passports issued per year. 2004-2006: There was an increase of 1.6 million passports issued per year. c.How are the different rates of change shown on the graph? Since the second rate of change is larger, the line there is steeper. 2002 2004 2006 Year Passports (millions) 7.0 8.9 12.1

7.  Choose the best answer for the following. A.The graph shows the number of airplane departures in the United States in recent years. Find the rates of change for 1995-2000 and 2000-2005. A.1,200,000 per year; 900,000 per year B.8,100,000 per year; 9,000,000 per year C.900,000 per year; 900,000 per year D.180,000 per year; 180,000 per year

8.  Choose the best answer for the following. B.Explain the meaning of the slope in each case. A.For 1995-2000, the number of airplane departures increased by about 900,000 flights each year. For 2000-2005, the number of airplane departures increased by about 180,000 flights each year. B.The rate of change was the same for 1995-2000 and 2000-2005. C.The number of airplane departures decreased by about 180,000 for 1995-2000 and 180,000 for 2000-2005. D.For 1995-2000 and 2000-2005, the number of airplane departures was the same.

9.  Choose the best answer for the following. C.How are the different rates of change shown on the graph? A.There is a greater vertical change for 1995-2000 than for 2000-2005. Therefore, the section of the graph for 1995-2000 has a steeper slope. B.They have different y-values. C.The vertical change for 1995-2000 is negative, and for 2000-2005 it is positive. D.The vertical change is the same for both periods, so the slopes are the same.

10.  A rate of change is constant for a function when the rate of change is the same between any pair of points on the graph of the function.  Linear functions have a constant rate of change.  A positive rate of change indicates an increase over time.  A negative rate of change indicates that a quantity is decreasing.

11.  Determine whether each function is linear. Explain. a.. b.. XY 16 212 318 424 +6 +1 Since the rate of change is constant, the function is linear. XY -105 -21 6-4 14-10 -4 -5 -6 +8 Since the rate of change is not constant, the function is not linear.

12.  Choose the best answer for the following. A.Determine whether the function is linear. Explain. A.Yes, the rate of change is constant. B.No, the rate of change is constant. C.Yes, the rate of change is not constant. D.No, the rate of change is not constant. XY 52 104 156 208

13.  Choose the best answer for the following. B.Determine whether the function is linear. Explain. A.Yes, the rate of change is constant. B.No, the rate of change is constant. C.Yes, the rate of change is not constant. D.No, the rate of change is not constant. XY 312 616 1220 1524

14.  The slope of a nonvertical line is the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run) as you move from one point to another.  It can be used to describe a rate of change.  Slope describes how steep a line is.  The greater the absolute value of the slope, the steeper the line.  Because a linear function has a constant rate of change, any two points on a nonvertical line can be used to determine its slope.

15.  The slope of a nonvertical line is the ratio of the rise to the run.  The slope m of a nonvertical line through any two points (x 1, y 1 ) and (x 2, y 2 ) can be found as follows.  The slope of a line can be positive, negative, zero, or undefined.  If the line is not horizontal or vertical, then the slope is either positive or negative.

16.  Find the slope of the line that passes through each pair of points. a.(-3, 2) and (5, 5) b.(-3, -4) and (-2, -8) c.(-3, 4) and (4, 4) x1x1 y1y1 y2y2 x2x2 x1x1 y1y1 y2y2 x2x2 x1x1 y1y1 y2y2 x2x2

17.  Choose the best answer for the following. A.Find the slope of the line that passes through (4, 5) and (7, 6). A.3 B.- 1 / 3 C. 1 / 3 D.-3

18.  Choose the best answer for the following. B.Find the slope of the line that passes through (-3, -5) and (-2, -7). A.2 B.-2 C.- 1 / 2 D. 1 / 2

19.  Choose the best answer for the following. C.Find the slope of the line that passes through (-3, -1) and (5, -1). A.Undefined B.8 C.2 D.0

20.  Find the slope of the line that passes through (-2, -4) and (-2, 3).

21.  Choose the best answer for the following. ◦ Find the slope of the line that passes through (5, -1) and (5, -3). A.Undefined B.0 C.4 D.2

22.  The graphs of lines with different slopes are summarized below. ◦ Positive Slope  Line slopes up from left to right. ◦ Negative Slope  Line slopes down from left to right. ◦ Slope of 0  Horizontal line. ◦ Undefined Slope  Vertical line.

23.  Sometimes you are given the slope and must find a missing coordinate.

24.  Find the value of r so that the line through (6, 3) and (r, 2) has a slope of ½. x1x1 y1y1 y2y2 x2x2 m = 1(r – 6) = -1(2) r – 6 = -2 +6 r = 4 m

25.  Choose the best answer for the following. ◦ Find the value of p so that the line through (p, 4) and (3, -1) has a slope of - 5 / 8. A.5 B.- 5 / 8 C.-5 D.11 m = -5(3 – p) = -5(8) +15 5p = -25 -15 – (-5p) = -40 -15 + 5p = -40 5 5