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§6.1 Rate of Change and Slope

§6.1 Rate of Change and Slope. Warm-Up. Evaluate each function rule for x = –5. 1. y = x – 7 2. y = 7 – x 3. y = 2 x + 5 4. y = – x + 3 Write in simplest form. 5. 6. 7. 2 5. 7 – 3 3 – 1. 3 – 5 6 – 0. 8 – (–4) 3 – 7. Warm-Up (Solutions).

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§6.1 Rate of Change and Slope

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  1. §6.1 Rate of Change and Slope

  2. Warm-Up Evaluate each function rule for x = –5. 1. y = x – 7 2.y = 7 – x 3.y = 2x + 5 4.y = – x + 3 Write in simplest form. 5.6.7. 2 5 7 – 3 3 – 1 3 – 5 6 – 0 8 – (–4) 3 – 7

  3. Warm-Up (Solutions) 1.y = x – 7 for x = –5: y = –5 – 7 = –12 2.y = 7 – x for x = –5: y = 7 – (–5) = 12 3.y = 2x + 5 for x = –5: y = 2(–5) + 5 = –10 + 5 = –5 4.y = – x + 3 for x = –5: y = – (–5) + 3 = 2 + 3 = 5 5. = = 2 6. = – = – 7. = – = –3 2 5 2 5 7 – 3 3 – 1 4 2 3 – 5 6 – 0 2 6 1 3 8 – (–4) 3 – 7 12 4

  4. Vocabulary Rate of Change: the relationship between two quantities that are changing. The rate of change is also called the slope. change in the dependent variable rate of change = change in the independent variable Slope: the ratio of the vertical change to the horizontal change. vertical changey2 – y1 slope = = , where x2 – x1 does not = 0. horizontal change x2 – x1

  5. For the data in the table, is the rate of change the same for each pair of consecutive mileage amounts? Fee for Miles Driven Miles Fee 100 $30 150 $42 200 $54 250 $66 Example 1: Finding Rate of ChangeUsing a Table Find the rate of change for each pair of consecutive mileage amounts.

  6. change in costCost depends on the change in number of milesnumber of miles. rate of change = Fee for Miles Driven 42 – 301254 – 421266 – 5412 150–100 50 200–150 50 200–250 50 = = = Miles Fee 100 $30 150 $42 200 $54 250 $66 Example 1: Finding Rate of ChangeUsing a Table (continued) The rate of change for each pair of consecutive mileage amounts is $12 per 50 miles. The rate of change is the same for all the data.

  7. Below is a graph of the distance traveled by a motorcycle from its starting point. Find the rate of change. Explain what this rate of change means. Choose two points On the graph (0,0) and (20,400) Example 2: Finding Rate of ChangeUsing a Graph

  8. vertical changechange in distance horizontalchangechange in time rate of change = 400 – 0 20 – 0 400 20 20 Use two points. = Divide the vertical change by the horizontal change. = Simplify. = Example 2: Finding Rate of ChangeUsing a Graph (continued) Using the points (0,0) and (20,400), find the rate of change. The rate of change is 20 m/s. The motorcycle is traveling 20 meters each second.

  9. a. rise run 4 – 1 0 – 2 slope = = 3 –2 3 2 = = – 3 2 The slope of the line is – . Example ¥: Finding SlopeUsing a Graph Find the slope of each line.

  10. Find the slope of the line. b. rise run 1 – -1 -1 – (-2) slope = = 2 1 = = 2 Example ¥: Finding SlopeUsing a Graph (continued) The slope of the line is 2.

  11. –1 – (–2) –2 – 3 = y2 – y1 x2 – x1 slope = Substitute (–2, –1) for (x2, y2) and (3, –2) for (x1, y1). Simplify. 1 –5 1 5 = = – 1 5 The slope of EF is – . Example 3: Finding SlopeUsing Points Find the slope of the line through E(3, –2) and F(–2, –1).

  12. a. y2–y1 x2 – x1 slope = 2 – 2 1– (–4) Substitute (1, 2) for (x2, y2) and (–4, 2) for (x1, y1). Simplify. = 0 5 = = 0 Example 4: Finding Slope ofHorizontal and Vertical Lines Find the slope of the line. The slope of all horizontal lines is 0.

  13. y2 – y1 x2 – x1 slope = Substitute (2, 1) for (x2, y2) and (2, –4) for (x1, y1). 1 – (–4) 2 – 2 = 5 0 = Simplify. Example 4: Finding Slope ofHorizontal and Vertical Lines (continued) Find the slope of the line. b. Division by zero is undefined. The slope of all vertical lines is undefined.

  14. Assignment: Pg. 298-299 8-37 Left OMIT 27, 29, 31, 35

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