Slope. Domain 2 Lesson 11. Getting the Idea

Transcription

1 Domain Lesson Slope Common Core Standard: 8.EE. Getting the Idea The graph of a linear equation is a straight line. The steepness of the line is called its slope. The slope shows the rate at which two quantities are changing. Specificall, it is the ratio of the vertical change to the horizontal change, or rise run. positive slope negative slope slope of 0 undefined slope Look at the lines graphed above. A line that slants up from left to right has a positive slope. For this graph, as the -values increase, the -values also increase. A line that slants down from left to right has a negative slope. For this graph, as the -values increase, the -values decrease. A horizontal line has a slope of 0 because there is no vertical change. A vertical line has an undefined slope, because there is no horizontal change. A fraction with a denominator of 0 is undefined, and rise run would have a denominator of 0. Duplicating an part of this book is prohibited b law. 00

2 Eample What is the slope of this line? Strateg Step Find the vertical change and the horizontal change. Choose two points on the line. (, ) and (, ) 0 Duplicating an part of this book is prohibited b law. Step Step Solution Identif the vertical change and horizontal change. From (, ) to (, ), the line moves units down and units to the right. Write the slope. The line slants down from left to right, so the slope is negative. slope rise run The slope is. (, ) 0 (, ) 0

3 Similar triangles have the same shape, but not necessaril the same size. The ratios of the lengths of their corresponding sides are equal. Eample Does the slope of a non-vertical line change depending on which two points ou use to determine it? Use the two similar triangles and the line graphed below to help ou answer the question (8, ) (, ) (, ) Strateg For each triangle, write the ratio of the vertical side length to the horizontal side length. Compare those ratios, and compare them to the slope of the line. Step Step Find the ratio of the vertical and horizontal side lengths for each triangle. vertical side length Smaller triangle: horizontal side length vertical side length Larger triangle: horizontal side length Simplif: Compare the ratios. The ratio of the lengths of the vertical and horizontal sides for both triangles is. Duplicating an part of this book is prohibited b law. 0 Domain : Epressions and Equations

4 Lesson : Slope Step Step Solution Compare the ratios to the slope. The slope is the ratio rise vertical change run or horizontal change. If ou use the points (, ) and (, ), slope rise run. If ou use the points (, ) and (8, ), slope = rise run. Analze the slope. No matter which two points on the line ou use, the slope is the same. This is because the graph of a line changes at a constant rate. The slope of a non-vertical line stas the same no matter which two points ou use to determine it. Eample demonstrates that the slope of a line represents a constant rate of change. If ou know an two points on a line, ou can determine its slope using the formula below. Slope The slope, m, of a line containing points (, ) and (, ) is: change in m change in Eample What is the slope of a line that passes through (, ) and (, )? Strateg Use the slope formula. Duplicating an part of this book is prohibited b law. Step Step Solution Identif the points. Let (, ) (, ). Let (, ) (, ). Substitute the numbers into the slope formula. m m m 8 m 8 () The slope is 8. 0

5 The slope of a line represents a constant rate of change. For eample, a slope of 0 could represent these rates: $0 hours 0 miles 0 pages gallons minutes When a graph shows a real-world situation, think about what the slope represents. Eample A locksmith charges a flat fee for each house call plus an hourl rate, as shown b the graph below. Total Charge (in dollars) House Call (, 70) (, 0) 7 8 Number of Hours Find the slope of the graph. What does the slope represent in this problem? Strateg Find and interpret the slope of the line. Step Step Solution Find the slope. Let (, ) (, 0). Let (, ) (, 70). m Interpret the slope. change in The slope shows the ratio change in. So, the slope compares the total charge, in dollars, to the number of hours worked. 0 dollars The slope shows a rate of change of, or $0 per hour. hour So, the slope represents the hourl rate charged b the locksmith. The slope is 0. It shows that the locksmith charges $0 per hour for each job. Duplicating an part of this book is prohibited b law. 0 Domain : Epressions and Equations

6 Lesson : Slope Coached Eample Joanie bought an airplane phone card that charges her a connection fee plus an additional rate for each minute a call lasts. The graph below represents this situation. Airplane Phone Card Costs Total Cost (in dollars) Number of Minutes What is the slope of the graph, and what does it represent? Choose an two points on the graph. Let (, ) (, ). Duplicating an part of this book is prohibited b law. Let (, ) (, ). m The -ais shows the total cost in. The -ais shows the time in. So, the slope shows a rate of change of dollars to, or per minute. Does the slope represent the connection fee or the rate per minute for the call? The slope is. It shows that Joanie must pa $ per minute for the calls she makes. 0

7 Lesson Practice Choose the correct answer.. What is the slope of this line? 0 A. C. _ B. _ D. Use the graph for questions and. A plumber charges a set fee for each house call plus an hourl rate, as shown b the graph below. Total Charge (in dollars) Plumbing Jobs Number of Hours. Which is true of this graph? 0 A. The slope is 0. B. The slope is. C. The slope is. D. The slope is undefined.. What is the slope of a line that passes through (, ) and (, )? _ A. C. _ B. _ D. _. What is the slope of the line graphed above? A. 00 B. 7 C. 0 D.. What does the slope of the graph represent? A. the cost of materials for each plumbing job B. the total charge for an plumbing job C. the set fee for an plumbing job D. the hourl rate charged b the plumber Duplicating an part of this book is prohibited b law. 0 Domain : Epressions and Equations

8 Lesson : Slope. What is the slope of this line? 7. Which is true of this graph? 0 0 A. C. _ B. _ D. A. The slope is. B. The slope is 0. C. The slope is. D. The slope is undefined. 8. Consider the line graphed below. 0 (, ) (, ) Duplicating an part of this book is prohibited b law. A. Using the two points labeled on the graph above, calculate its slope. Show our work. B. Does the slope of the line change depending on which two points ou use? Calculate the slope using two different points to support our answer. 07

9 9. Circle the number that represents the slope of the graph shown below. 0 The slope of the graph is. 0. Find the slope of the line that connects each point below and (, ). Write the coordinates of the point in the correct bo. (, ) (, ) (, ) (, ) (, ) (, ) Slope, 0 Slope 0 Slope. 0. Select True or False for each statement about the graph shown below. 0 A. The slope is negative. True False B. The slope is positive. True False C. The slope is 0. True False D. The line slants down from left to right. True False E. The line slants up from left to right. True False Duplicating an part of this book is prohibited b law. 08 Domain : Epressions and Equations

10 Lesson : Slope. An electrician charges a set fee for each house call plus an hourl rate, as shown in the graph below. Circle ever statement that correctl describes the information represented b the graph. Total Cost (in dollars) Electrician Fees 7 8 Number of House Calls A. The hourl rate is $0. B. The hourl rate is $0. C. The hourl rate is $0. D. The set fee is $0. E. The maimum amount the electrician charges is $0. F. If the electrician works hours, the total cost is $0.. Draw a line from each pair of points to the slope of the line that connects those points. A. (, ), (8, ) B. (9, 7), (9, ) _ C. (9, ), (, 0) 0 Duplicating an part of this book is prohibited b law. D. (, ), (, ). Use numbers from the bo to complete the statements shown below. The line through (, ) and (, ) has a slope of. The line through (, ) and (, ) has a slope of. The line through (, ) and (, ) has a slope of. 8 09