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7-7 Finding Slope of a Line Preview Warm Up California Standards Lesson Presentation Holt CA Course 1 7-7 Finding Slope of a Line Warm Up Evaluate each equation for x = –1, 0, and 1. 1. y = 3x –3, 0, 3 2. y = x – 7 –8, –7, –6 3. y = 2x + 5 3, 5, 7 4. y = 6x – 2 –8, –2, 4 Holt CA Course 1 7-7 Finding Slope of a Line California Standards AF3.4 Plot the values of quantities whose ratios are always the same (e.g. cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities. Also covered: AF3.3. Holt CA Course 1 7-7 Finding Slope of a Line Recall that lines have constant slope. For a line on the coordinate plane, slope is the following ratio: vertical change change in y = horizontal change change in x Holt CA Course 1 7-7 Finding Slope of a Line If you know any two points on a line, you can find the slope of the line without graphing. The slope of a line through the points (x1, y1) and (x2, y2) is as follows: y2 – y1 slope = x 2 – x1 When finding slope using the ratio above, it does not matter which point you choose for (x1, y1) and which point you choose for (x2, y2). Holt CA Course 1 7-7 Finding Slope of a Line Additional Example 1: Finding Slope, Given Two Points Find the slope of the line that passes through A. (–2, –3) and (4, 6). Let (x1, y1) be (–2, –3) and (x2, y2) be (4, 6). y2 – y1 6 – (–3) Substitute 6 for y2, –3 for y1, x2 – x1 = 4 – (–2) 4 for x2, and –2 for x1. = 6 + 2 =9 3 4+ 6 =3 Simplify. 2 The slope of the line that passes through (–2, –3) and (4, 6) is 3. 2 Holt CA Course 1 7-7 Finding Slope of a Line Additional Example 1: Finding Slope, Given Two Points Find the slope of the line that passes through B. (1, 3) and (2, 1). Let (x1, y1) be (1, 3) and (x2, y2) be (2, 1). y2 – y1 1 – 3 Substitute 1 for y2, 3 for y1, x2 – x1 = 2 – 1 2 for x2, and 1 for x1. -2 Simplify. = 1 = –2 The slope of the line that passes through (1, 3) and (2, 1) is –2. Holt CA Course 1 7-7 Finding Slope of a Line Additional Example 1: Finding Slope, Given Two Points Find the slope of the line that passes through C. (3, –2) and (1, –2). Let (x1, y1) be (3, –2) and (x2, y2) be (1, –2). y2 – y1 –2 – (–2) Substitute -2 for y2, -2 for y1, x2 – x1 = 1 – 3 1 for x2, and 3 for x1. -2 + 2 Rewrite subtraction as addition of =1–3 the opposite. 0 = –2 = 0 The slope of the line that passes through (3, –2) and (1, –2) is 0. Holt CA Course 1 7-7 Finding Slope of a Line Check It Out! Example 1 Find the slope of the line that passes through A. (–4, –6) and (2, 3). Let (x1, y1) be (–4, –6) and (x2, y2) be (2, 3). y2 – y1 3 – (–6) Substitute 3 for y2, –6 for y1, x2 – x1 = 2 – (–4) 2 for x2, and –4 for x1. =9 =3 6 2 The slope of the line that passes through (–4, –6) and (2, 3) is 3. 2 Holt CA Course 1 7-7 Finding Slope of a Line Check It Out! Example 1 Find the slope of the line that passes through B. (2, 4) and (3, 1). Let (x1, y1) be (2, 4) and (x2, y2) be (3, 1). y2 – y1 1 – 4 Substitute 1 for y2, 4 for y1, x2 – x1 = 3 – 2 3 for x2, and 2 for x1. -3 Simplify. = 1 = –3 The slope of the line that passes through (2, 4) and (3, 1) is –3. Holt CA Course 1 7-7 Finding Slope of a Line Check It Out! Example 1 Find the slope of the line that passes through C. (3, –2) and (1, –4). Let (x1, y1) be (3, –2) and (x2, y2) be (1, –4). y2 – y1 –4 – (–2) Substitute -4 for y2, -2 for y1, x2 – x1 = 1 – 3 1 for x2, and 3 for x1. = –4 + 3 1– 2 = –2 = 1 Simplify. –2 The slope of the line that passes through (3, –2) and (1, –4) is –2. Holt CA Course 1 7-7 Finding Slope of a Line Additional Example 2: Money Application The table shows the total cost of fruit per pound purchased at the grocery store. Use the data to make a graph. Find the slope of the line and explain what it shows. Cost of Fruit Graph the data. Cost Pounds Holt CA Course 1 7-7 Finding Slope of a Line Helpful Hint You can use any two points to find the slope of the line. Holt CA Course 1 7-7 Finding Slope of a Line Additional Example 2 Continued Cost of Fruit Find the slope of the line: y2 – y1 x2 – x1 Cost 30 - 15 Substitute. 10 - 5 15 = 3 Multiply. 5 Pounds The slope of the line is 3. This means that for every pound of fruit, you will pay another $3. Holt CA Course 1 7-7 Finding Slope of a Line Check It Out! Example 2 The table shows the total cost of gas per gallon. Use the data to make a graph. Find the slope of the line and explain what it shows. Cost of Gas Graph the data. y Cost of Gas 12 Gallons Cost Cost 9 0 0 6 3 6 3 6 12 0 3 6 9 x Gallons Holt CA Course 1 7-7 Finding Slope of a Line Check It Out! Example 2 Continued Cost of Gas Find the slope of the line: y y2 – y1 12 x2 – x1 = Cost 9 12 - 6 Substitute. 6-3 6 6 = 2 Multiply. 3 3 0 3 6 9 x Gallons The slope of the line is 2. This means that for every gallon of gas, you will pay another $2. Holt CA Course 1 7-7 Finding Slope of a Line The slope of a line may be positive, negative, zero, or undefined. You can tell which of these is the case by looking at the graphs of a line— you do not need to calculate the slope. Holt CA Course 1 7-7 Finding Slope of a Line Holt CA Course 1 7-7 Finding Slope of a Line Holt CA Course 1 7-7 Finding Slope of a Line Lesson Quiz: Part I Find the slope of the line that passes through each pair of points. 1. (4, 3) and (–1, 1) 2 5 2. (–1, 5) and (4, 2) –3 5 Holt CA Course 1 7-7 Finding Slope of a Line Lesson Quiz: Part II 3. The table shows how much money Susan earned as a house painter for one afternoon. Use the data to make a graph. Find the slope of the line and explain what it shows. y The slope of the line is 7. This means Susan earned 80 $7 for each hour worked. 70 60 50 40 30 20 10 x 0 2 4 6 8 10 12 14 Holt CA Course 1