6.3 Using Slope LESSON EXPLORE ACTIVITY 1. rate of change =

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1 v? LESSON 6.3 Using Slope ESSENTIL QUESTION Calculate and interpret the average rate of change of a function (presented smbolicall or as a table) over a specified interval. Estimate the rate of change from a graph. lso F.IF.4, F.IF.7, F.LE.1b How can ou relate rate of change and slope in linear relationships? EXPLORE CTIVITY 1 Finding Rates of Change For a function defined in terms of and, the rate of change over a part of the domain of the function is a ratio that compares the change in to the change in in that part of the domain. rate of change = change in change in In 004, the cost of sending a 1ounce letter was 37 cents. In 008, the cost was 4 cents. Find the rate of change in cost for this time period. rate of change = change in change in = = = The rate of change was cents per ear. Find the rate of change for each time period. Year () Cost in cents () Houghton Mifflin Harcourt Publishing Compan 1988 to 1990 = = cents per ear to 1991 = = cents per ear to 004 = = = cents per ear Round to the nearest hundredth of a cent. 004 to 008 = = = cents per ear Lesson

2 EXPLORE CTIVITY 1 (cont d) 1. Interpret the nswer The rate of change for 004 to 008 was 1.5 cents per ear. Does this mean the actual change in cost each ear was 1.5 cents? Eplain. Finding Slope of a Line In the previous Eplore ctivit, the rate of change was not constant. It varied from 0 to 4 cents per ear. However, for linear functions, the rate of change is constant. The rate of change for a linear function can be calculated using the rise and run of the graph of the function. The rise is the difference in the values of two points on a line. The run is the difference in the values of two points on a line. The slope of a line is the ratio of rise to run for an two points on the line. 6 4 O Slope = run difference in values = difference in values EXMPLE 1 M Notes Find the slope of each line. 3 (, 3) 1 (1, 1) O (1, 4) O (3, 0) 4 Use (, 3) as the starting point. Subtract values to find the change in or rise. Then subtract values to find the change in or run. slope = = 1 = It doesn t matter which point ou start with as long as ou are consistent. If ou start with (1, 1), then slope = = 1 =. Use ( 1, 4) as the starting point. Subtract values to find the change in or rise. Then subtract values to find the change in or run. 0 4 slope = 4 = 1 3(1) = 4 Houghton Mifflin Harcourt Publishing Compan 17 Unit

3 YOUR TURN. Find the slope of the line in the graph. 5 3 O 5 5 EXPLORE CTIVITY Classifing Slopes s shown in the previous eample, slope can be positive or negative. What about the slope of horizontal and vertical lines? Find the slope of each line. (1, ) (3, ) O run = = = Houghton Mifflin Harcourt Publishing Compan (, 3) (, 1) 3 3 O 3 3 run = = 0 Since ou cannot divide b 0, the slope is undefined. Lesson

4 EXPLORE CTIVITY (cont d) Positive Slope Negative Slope Zero Slope Undefined Slope Line rises from left to right. Line falls from left to right. Horizontal line Vertical line 3. Communicate Mathematical Ideas Eplain wh the slope of a vertical line is undefined. Using the Slope Formula The slope formula for the slope of a line is the ratio of the difference in values to the difference in values between an two different points on the line. This means if ( 1, 1 ) and (, ) are an two points on a line, the slope is m = 1. 1 EXMPLE Find the slope of the line passing through the points (5, 3) and ( 1, 15). Describe the slope as positive, negative, zero, or undefined. nimated Math STEP 1 Find the rise or difference in values. STEP Find the run or difference in values. 1 = 15 3 = 1 1 = 1 5 = 6 STEP 3 Find the slope. STEP 4 Describe the slope. YOUR TURN run = 1 6 = The slope is negative. The line falls from left to right. 4. Find the slope of the line passing through the points (9, 1) and (1, 4). Describe the slope as positive, negative, zero, or undefined. Houghton Mifflin Harcourt Publishing Compan 174 Unit

5 Interpreting Slope EXMPLE 3 CORE F.IF.4, F.IF.6, F.IF.7, F.LE.1b The graph shows the relationship between a person s age and his or her estimated maimum heart rate. Maimum heart rate (beats/min) Estimated Maimum Heart Rate (0, 00) (60, 160) O ge (r) Find the slope. Interpret the slope. Use the two points that are labeled on the graph. run = 60 0 = = 1 The slope of 1 means that for ever ear a person s age increases, his or her maimum heart rate decreases b 1 beat per minute. 5. MultiStep Tara and Jade are hiking up a hill together. Each has a different stride. The run for Tara s stride is 3 inches, and the rise is 8 inches. The run for Jade s stride is 36 inches. What is the rise of Jade s stride? What is the slope and what does it mean in this problem? Math Talk Mathematical Practices Wh is it important to know both the formula and the description of what slope is? Houghton Mifflin Harcourt Publishing Compan YOUR TURN 6. In an eperiment, a car began traveling at a constant speed at 1:00 PM. Over a period of 5 hours, the car traveled a total 160 miles graph shows the relationship between the length of time that the car had been traveling since 1:00 PM and the number of miles that it had traveled. What is the slope of the line? What does the slope mean? Lesson