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Section 3.4

Section 3.4. Slope and Rates of Change. Page 190. Slope. The rise , or change in y, is y 2  y 1 , and the run , or change in x, is x 2 – x 1. Example. Page 191. Use the two points to find the slope of the line. Interpret the slope in terms of rise and run. Solution. ( –4 , 1 ).

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Section 3.4

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  1. Section 3.4 • Slope and Rates of Change

  2. Page 190 Slope • The rise, or change in y, is y2y1, and the run, or change in x, is x2 – x1.

  3. Example Page 191 • Use the two points to find the slope of the line. Interpret the slope in terms of rise and run. • Solution (–4, 1) (0, –2) The rise is 3 units and the run is –4 units.

  4. Example Page 192 • Calculate the slope of the line passing through each pair of points. • a. (3, 3), (0, 4) b. (3, 4), (3, 2) • c. (2, 4), (2, 4) d. (4, 5), (4, 2) • Solution

  5. Example Page 192 • Calculate the slope of the line passing through each pair of points. • a. (3, 3), (0, 4) b. (3, 4), (3, 2) • c. (2, 4), (2, 4) d. (4, 5), (4, 2) • Solution

  6. Example Page 192 • Calculate the slope of the line passing through each pair of points. • a. (3, 3), (0, 4) b. (3, 4), (3, 2) • c. (2, 4), (2, 4) d. (4, 5), (4, 2) • Solution

  7. Example Page 192 • Calculate the slope of the line passing through each pair of points. • a. (3, 3), (0, 4) b. (3, 4), (3, 2) • c. (2, 4), (2, 4) d. (4, 5), (4, 2) • Solution

  8. Finding Slope of a Line, p 249 Find the slope of the line containing the points (-3, 4) and (-4,- 2) Find the slope of the line containing the points (4,-2) and (-1,5)

  9. Page 193 Slope Positive slope: rises from left to right Negative slope: falls from left to right

  10. Page 193 Slope Zero slope:horizontal line Undefined slope: vertical line

  11. Example Page 193 • Find the slope of each line. • a. b. • Solution • a. The graph rises 2 units for each unit of run m = 2/1 = 2. • b. The line is vertical, so the slope is undefined.

  12. Example Page 193 • Sketch a line passing through the point (1, 2) and having slope 3/4. • Solution • Start by plotting (1, 2). • The slope is ¾ which means a rise (increase) of 3 and a run (horizontal) of 4. • The line passes through the point (1 + 4, 2 + 3) = (5, 5).

  13. Page 195 Slope as a Rate of Change When lines are used to model physical quantities in applications, their slopes provide important information. Slope measures the rate of change in a quantity.

  14. Example Page 195similar to Example 7&8and #87 from homeworkand #91 • The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. • a. Find the y-intercept. What does the y-intercept represent? • Solution • a. The y-intercept is 35, so the boat is initially 35 miles from the dock.

  15. Example (cont) Page 195similar to Example 7&8and #87 from homeworkand #91 • The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. • b. The graph passes through the point (4, 15). Discuss the meaning of this point. • Solution • b. The point (4, 15) means that after 4 hours the boat is 15 miles from the dock.

  16. Example (cont) Page 195similar to Example 7&8and #87 from homeworkand #91 • The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. • c. Find the slope of the line. Interpret the slope as a rate of change. • Solution • c. The slope is –5. The slope means that the boat is going toward the dock at 5 miles per hour.

  17. Example: #88 p 202 • Electricity: The graph shows how voltage is related to amperage in an electrical circuit. The slope corresponds to the resistance in ohms. Find the resistance in this electrical circuit. • Find the slope of the line passing through the points. Look at the graph on page 202 and identify two points. • (0,0), (10, 20) and (20, 40) are possible • Interpret the slope as resistance in this electrical circuit. • 0.5 ohm

  18. Example: #91 p 202 • Median Household Income: In 2000, median family income was about $42,000, and in 2008 it was about $50,000. • Find the slope of the line passing through the points (2000,42000) and (2008,50000) • Interpret the slope as rate of change. • Median family income increased on average by $1000/year over this time period • If this trend continues, estimate the median family income in 2014. $56,000 ($14000 added to $42000 or $6000 added to $50,000

  19. Example • When a street vendor sells 40 tacos, his profit is $24, and when he sells 75 tacos, his profit is $66. • a. Find the slope of the line passing through the points (40, 24) and (75, 66) • b. Interpret the slope as a rate of change. • Solution • b. Profit increases on average, by $1.20 for each additional taco sold.

  20. Example #86 on page 202 • Profit from Tablet Computers: When a company manufactures 500 tablet computers, its profit is $100,000, and when it manufactures 1500 tablet computers, its profit is $400,000. • Find the slope of the line passing through the points (500, 100000) and (1500, 400000) • b. Interpret the slope as a rate of change. • The average profit is $300/tablets computer.

  21. DONE

  22. Objectives • Finding Slopes of Lines • Slope as a Rate of Change

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