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Preview Warm Up California Standards Lesson Presentation

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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

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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.

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**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 =

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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).

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**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

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**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.

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**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.

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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

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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.

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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.

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**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.

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

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**Additional Example 2 Continued**

Pounds Cost Cost of Fruit Find the slope of the line: y2 – y1 x2 – x1 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.

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**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

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**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.

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**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.

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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 –

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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.

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