Preview Warm Up California Standards Lesson Presentation

Warm Up Evaluate each equation for x = –1, 0, and 1. 1. y = 3x 2. y = x – 7 3. y = 2x + 5 4. y = 6x – 2 –3, 0, 3 –8, –7, –6 3, 5, 7 –8, –2, 4

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.

vertical change horizontal change Recall that lines have constant slope. For a line on the coordinate plane, slope is the following ratio: vertical change horizontal change change in y change in x =

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 x2 – x1 slope = 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).

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 x2 – x1 6 – (–3) 4 – (–2) Substitute 6 for y2, –3 for y1, 4 for x2, and –2 for x1. 6 + 3 4 + 2 = 9 6 = 3 2 = Simplify. The slope of the line that passes through (–2, –3) and (4, 6) is . 3 2

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 x2 – x1 1 – 3 2 – 1 Substitute 1 for y2, 3 for y1, 2 for x2, and 1 for x1. -2 1 = = –2 Simplify. The slope of the line that passes through (1, 3) and (2, 1) is –2.

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 x2 – x1 –2 – (–2) 1 – 3 Substitute -2 for y2, -2 for y1, 1 for x2, and 3 for x1. -2 + 2 1 – 3 = Rewrite subtraction as addition of the opposite. = 0 –2 = The slope of the line that passes through (3, –2) and (1, –2) is 0.

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 x2 – x1 3 – (–6) 2 – (–4) Substitute 3 for y2, –6 for y1, 2 for x2, and –4 for x1. 9 6 = 3 2 = The slope of the line that passes through (–4, –6) and (2, 3) is . 3 2

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 x2 – x1 1 – 4 3 – 2 Substitute 1 for y2, 4 for y1, 3 for x2, and 2 for x1. -3 1 = = –3 Simplify. The slope of the line that passes through (2, 4) and (3, 1) is –3.

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 x2 – x1 –4 – (–2) 1 – 3 Substitute -4 for y2, -2 for y1, 1 for x2, and 3 for x1. –4 + 2 1 – 3 = = 1 –2 = Simplify. The slope of the line that passes through (3, –2) and (1, –4) is –2.

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. Pounds Cost Cost of Fruit Graph the data.

Helpful Hint You can use any two points to find the slope of the line.

Additional Example 2 Continued Pounds Cost Cost of Fruit Find the slope of the line: y2 – y1 x2 – x1 30 - 15 10 - 5 Substitute. 15 5 = 3 Multiply. The slope of the line is 3. This means that for every pound of fruit, you will pay another $3.

Cost of Gas Gallons Cost 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. 6 9 12 3 x y Gallons Cost of Gas Cost Graph the data. Cost of Gas Gallons Cost 3 6 12

Check It Out! Example 2 Continued 6 9 12 3 x y Gallons Cost of Gas Cost Find the slope of the line: = y2 – y1 x2 – x1 12 - 6 6 - 3 Substitute. 6 3 = 2 Multiply. The slope of the line is 2. This means that for every gallon of gas, you will pay another $2.

The slope of a line may be positive, negative, zero, or undefined 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.

Lesson Quiz: Part I Find the slope of the line that passes through each pair of points. 1. (4, 3) and (–1, 1) 2. (–1, 5) and (4, 2) 2 5 5 3 –

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. x y 6 4 2 8 10 12 14 20 30 40 50 60 70 80 The slope of the line is 7. This means Susan earned $7 for each hour worked.