Rate of Change and Slope 3.2

Definitions Rate of Change: The ratio of the amount of change in the dependent variable (output) to the amount of change in the independent variable (input) Input: the x value of the table (the independent variable) Output: The y value of the table ( the dependent variable) Constant: Is something or a number that happens ALL the time (constantly) Variable: Is something or a number that does not happen very much or at all

Example 1: John keeps a record of the number of lawns he has mowed and the money he has earned. Tell whether the rates of change are constant or a variable. Day 1 Day 2 Day 3 Day 4 Number of lawns 1 3 6 8 Amount earned ($) 15 45 90 120 Step 2: Find the rates of change Day 1 to Day 2 Change in $ 45-15 = 30 =15 Change in lawns 3-1 2 Day 2 to Day 3 Change in $ 90-45 = 45 =15 Change in lawns 6-3 3 Day 3 to Day 4 Change in $ 120-90 = 30 = 15 Change in lawns 8-6 2 Step 1: Identify the input and output variables Number of lawn (input) Amount earned (output) Step 3: The rate of change is $15 per lawn… it is a constant rate of change

Additional Example: Hector keeps a record of the total number of clients he has an the amount he earns as a personal trainer. Tell whether the rates are constant or variable. Step 1: Identify the input and output variables Number of clients (input) Amount earned (output Step 2: Find the rates of change Day 1 to Day 2 Change in $ 135-45 = 90 =45 Change in clients 3-1 2 Step 3: The rate of change is $45 per client… it is a constant rate of change Day 1 Day 2 Day 3 Day 4 # of clients 1 3 4 7 $ earned 45 135 180 315 Day 2 to Day 3 Change in $ 180-135 = 45 =45 Change in clients 4-3 1 Day 3 to Day 4Change in $ 315-180 = 135 = 45 Change in clients 7-4 3

Page 77 # 1 Complete page 78 with a partner

Calculating Slope Definitions: Slope of a line is the ratio of the change in y-value (rise) for a segment on the graph to the corresponding change in the x-values (run) Slope is known as the variable “m”

Why learn slope? Slope is very important in the construction field because it often dictates the best way to complete a project. Special building materials are used for low-slope roofs, so a construction worker must know the slope of the roof before beginning a shingling project. The slope of a road affects water runoff, so civil engineers and construction workers must plan accordingly. Most municipalities have rules regarding the minimum slope of their roads. Skiing enthusiasts pay special attention to the slopes of their favorite trails. The greater the slope of a trail, the more challenging it is. Knowing the slope of a particular trail makes it easier for skiers to judge the danger of skiing on that trail. Slope is also important to people who use wheelchairs. If a wheelchair ramp is too steep, it may be difficult to use. Constructing a ramp with the right slope is especially important for people who use manual wheelchairs instead of power wheelchairs. If a ramp is too steep, the user may not have the upper arm strength necessary to power a manual chair up the ramp

Guess the slope?

Finding the slope given two points: Find the slope of the line that passes through (2,3) and (4,-1) Two ways to do this: With a picture Rise = -4 = -2 m = -2 Run 2 y2-y1 -1 -3 = -4 = -2 m = -2 x2-x1 4-2 2 Find the slope of the line that passes through (2,3) and (4,-1) Two ways to do this: With a picture With a formula

Another Example: Find the slope through (3,2) and (-1,5) With a picture With a formula y2-y1 5-2 = 3= m = −3 4 x2-x1 -1-3 -4