Rate of Change and Slope 3-3 Notes for Algebra 1 Rate of Change and Slope

Rate of Change A ratio that describes how much one quantity changes with respect to a change in another quantity. x—is the independent variable. y—is the dependent variable. 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒= 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 A positive rate of change indicates an increase over time. A negative rate of change indicates that a quantity is decreasing.

Example 1 pg. 172 Find the Rate of Change DRIVING TIME Use the table to find the rate of change. Then explain its meaning. Time Driving (h) Distance Traveled (mi) x Y 2 76 4 152 6 228

Example 1 pg. 172 Find the Rate of Change DRIVING TIME Use the table to find the rate of change. Then explain its meaning. 38 1 ; this means the car is traveling at a rate of 38 miles per hour. Time Driving (h) Distance Traveled (mi) x Y 2 76 4 152 6 228

Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 1.) Find the rate of change for 2002-2004 9 and 2004-2006. 7 2.) Explain the meaning of the rate of change 5 in each case. 0 2002 2004 2006 3.) How are the different rates of change shown on the graph. 12.1 8.9 7.0

Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 1.) Find the rate of change for 2002-2004 9 and 2004-2006. 7 5 0 2002 2004 2006 950,000/yr 1,600,000/yr 12.1 8.9 7.0

Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 9 7 2.) Explain the meaning of the rate of change 5 in each case. 0 2002 2004 2006 For 2002-2004 there was an annual increase of 950,000 passports issued. Between 2004-2006, there was an average yearly increase of 1,600,000 passports issued. 12.1 8.9 7.0

Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 9 7 5 0 2002 2004 2006 3.) How are the different rates of change shown on the graph. There is a greater vertical change for 2004-2006 than for 2002-2004. Therefore, the section of the graph for 2004-2006 is steeper. 12.1 8.9 7.0

Slope (m) The slope of a non-vertical line is the ratio of the change in the y- coordinates (rise) to the change in the x-coordinates (run) 𝑠𝑙𝑜𝑝𝑒= 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 𝑥 1 , 𝑦 1 , 𝑥 2 , 𝑦 2 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦−𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥−𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1

Example 4 pg. 175 Positive, Negative and Zero slopes Find the slope of the line that passes through each pair of points. 1.) −3, 2 and 5, 5 2.) −3, −4 and −2, −8 3.) −3, 4 and 4, 4

Example 4 pg. 175 Positive, Negative and Zero slopes Find the slope of the line that passes through each pair of points. 1.) −3, 2 and 5, 5 𝑚= 3 8 2.) −3, −4 and −2, −8 𝑚=−4 3.) −3, 4 and 4, 4 𝑚=0

Example 5 pg. 176 Undefined Slope Find the slope of the line that passes through −2, −4 and −2, 3 .

Example 5 pg. 176 Undefined Slope Find the slope of the line that passes through −2, −4 and −2, 3 . Undefined

Slope Summary Positive Slope– goes uphill from left to right Negative Slope– goes downhill from left to right Slope of 0– horizontal line (y = #) Undefined Slope– vertical line (x = #)

Example 6 pg. 176 Find Coordinates given the slope. Find the value of r so that the line through 6, 3 and 𝑟, 2 has a slope of 1 2 .

Example 6 pg. 176 Find Coordinates given the slope. Find the value of r so that the line through 6, 3 and 𝑟, 2 has a slope of 1 2 . 𝑟=4

3-3 pg. 177 15-39o, 43-45, 54-66(x3)