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Algebra1 Rate of Change and Slope

Algebra1 Rate of Change and Slope

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Algebra1 Rate of Change and Slope

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  1. Algebra1Rate of Changeand Slope CONFIDENTIAL

  2. Warm Up Find the domain and range of each relation, and tell whether the relation is a function. 1) {(3, 4) , (3, 2) , (3, 0) , (3, -2)} 2) 1) D: (3) R: (4, 2, 0, -2) not a function 2) D: (0, 2, 4, -2, -4) R: (0, 2, 4, 2, 4) Relation is a function. CONFIDENTIAL

  3. Rate of Change and Slope A rate of change is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. Rate of change = Change in dependent variable Change in independent variable Rates of change can be used to find how quickly costs have increased. CONFIDENTIAL

  4. Consumer Application The table shows the cost of mailing a 1-ounce letter in different years. Find the rate of change in cost for each time interval. During which time interval did the cost increase at the greatest rate? Step1: Identify the dependent and independent variables. dependent: cost independent: year Step2: Find the rates of change. change in cost = _25 -22_ = 3 = 1 change in years 1988 – 1985 3 1985 to 1988 1 cent/year Next slide  CONFIDENTIAL

  5. change in cost = _25 -25_ = 0 = 0 change in years 1990 – 1988 2 1988 to 1990 0 cent/year change in cost = _29 -25_ = 4 = 4 change in years 1991 – 1990 1 1990 to 1991 4 cents/year change in cost = _37 -29_ = 8 = 0 change in years 2004 – 1991 13 1991 to 2004 8/13 cents/year The cost increased at the greatest rate from 1990 to 1991. CONFIDENTIAL

  6. Now you try! 1) The table shows the balance of a bank account on different days of the month. Find the rate of change during each time interval. During which time interval did the balance decrease at the greatest rate? 1) 1 to 6 = 53 dollars/day decrease 6 to 16 = 7.5 dollars/day decrease 16 to 22 = 0 dollars/day decrease 22 to 30 = 4.375 dollars/day decrease During time interval 1 to 6 the balance decreased at the greatest rate. CONFIDENTIAL

  7. Finding Rates of Change from a Graph 2) Graph the data from Example 1 and show the rates of change. Graph the ordered pairs. The vertical blue segments show the changes in the dependent variable, and the horizontal green segments show the changes in the independent variable. Notice that the greatest rate of change is represented by the steepest of the red line segments. Also notice that between 1988 and 1990, when the cost did not change, the red line segment is horizontal. CONFIDENTIAL

  8. Now you try! 2) Graph the data from the given table and show the rates of change. 2) The x-intercept is -2. The y-intercept is 3. CONFIDENTIAL

  9. Slope of a Line If all of the segments have the same rate of change, then they all have the same steepness and together form a straight line. The constant rate of change of a line is called the slope of the line. The rise is the difference in the y-values of two points on a line. The run is the difference in the x-values of two points on a line. The slope of a line is the ratio of rise to run for any two points on the line. slope = rise = Change in y run Change in x CONFIDENTIAL Next slide 

  10. (Remember that yis the dependent variableand xis the independent variable.) CONFIDENTIAL

  11. Finding Slope 3) Find the slope of the line. Begin at one point and count vertically to find the rise. Then count horizontally to the second point to find the run. It does not matter which point you start with. The slope is the same. slope = 2 = 2 1 slope = -2 = 2 -1 CONFIDENTIAL

  12. Now you try! 3) Find the slope of the line that contains (0, -3) and (5, -5) . 3) slope = -5 2 CONFIDENTIAL

  13. Finding Slopes of Horizontal and Vertical Lines 4) Find the slope of each line. A) B) slope = rise = 0 = 0 run 4 slope = rise = 2 run 0 You cannot divide by 0. The slope is 0. The slope is undefined. CONFIDENTIAL

  14. Now you try! 4) Find the slope of each line. A) B) 4a) slope = -3 – 5 . The slope is undefined. 0 - 0 4b) slope = -4 – (-4) . The slope is 0. 4 - 0 CONFIDENTIAL

  15. As shown in the previous examples, slope can be positive, negative, zero, or undefined. You can tell which of these is the case by looking at the graph of a line—you do not need to calculate the slope. CONFIDENTIAL

  16. Describing Slope 5) Tell whether the slope of each line is positive, negative, zero, or undefined. A) B) The line falls from left to right. The slope is negative. The line is horizontal. The slope is 0. CONFIDENTIAL

  17. Now you try! 5) Tell whether the slope of each line is positive, negative, zero, or undefined. A) B) 5a) The line is vertical. undefined slope. 5b) The rises from left to right. The slope is positive. CONFIDENTIAL

  18. The line with slope 4 is steeper than the line with slope 1 | 4 | > 1 2 2 Comparing Slope Remember that the slope of a line is its steepness. Some lines are steeper than others. As the absolute value of the slope increases, the line becomes steeper. As the absolute value of the slope decreases, the line becomes less steep. CONFIDENTIAL

  19. The line with slope -3 is steeper than the line with slope 3 | -3 | >| 3 | 4 | 4 | The line with slope -2 is steeper than the line with slope -1. | -2 | > | -1 | CONFIDENTIAL

  20. Now you try! 6) What is the rise shown in the graph? What is the run? What is the slope? 6) rise = 6; run = 5; slope = 6/5 CONFIDENTIAL

  21. Assessment 1) The table shows the volume of gasoline in a gas tank at different times. Find the rate of change for each time interval. During which time interval did the volume decrease at the greatest rate? 2) The table shows a person’s heart rate over time. Graph the data and show the rates of change. CONFIDENTIAL

  22. 150 125 100 75 50 25 0 2 4 6 8 10 12 1) 0 to 1 = 3 gallons/hr decrease 1 to 3 = 2 gallons/hrdecrease 3 to 6 = 4/3 gallons/hrdecrease 6 to 7 = 0 gallons/hrdecrease During time interval 0 to 1 the volume decreased at the greatest rate. 2) 1) 0 to 2 = 14 beats/min2 increase 2 to 5 = 18 beats/min2increase 5 to 6 = 20.3 beats/min2decrease 7 to 10 = 6.3 beats/min2decrease CONFIDENTIAL

  23. Find the slope of each line. 3) 4) 3) slope = 0.5 4) slope = 0 CONFIDENTIAL

  24. Tell whether the slope of each line is positive, negative, zero, or undefined. 6) 5) 5) The line falls from left to right. The slope is negative. 6) The line is vertical. Undefined slope. CONFIDENTIAL

  25. 7) The slope of which line has the greatest absolute value? 8) For which line is the run equal to 0? 7) line C 8) line D CONFIDENTIAL

  26. Let’s review Let’s review Rate of Change and Slope A rate of change is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. Rate of change = Change in dependent variable Change in independent variable Rates of change can be used to find how quickly costs have increased. CONFIDENTIAL

  27. Consumer Application The table shows the cost of mailing a 1-ounce letter in different years. Find the rate of change in cost for each time interval. During which time interval did the cost increase at the greatest rate? Step1: Identify the dependent and independent variables. dependent: cost independent: year Step2: Find the rates of change. change in cost = _25 -22_ = 3 = 1 change in years 1988 – 1985 3 1985 to 1988 1 cent/year Next slide  CONFIDENTIAL

  28. change in cost = _25 -25_ = 0 = 0 change in years 1990 – 1988 2 1988 to 1990 0 cent/year change in cost = _29 -25_ = 4 = 4 change in years 1991 – 1990 1 1990 to 1991 4 cents/year change in cost = _37 -29_ = 8 = 0 change in years 2004 – 1991 13 1991 to 2004 8/13 cents/year The cost increased at the greatest rate from 1990 to 1991. CONFIDENTIAL

  29. Finding Rates of Change from a Graph 2) Graph the data from Example 1 and show the rates of change. Graph the ordered pairs. The vertical blue segments show the changes in the dependent variable, and the horizontal green segments show the changes in the independent variable. Notice that the greatest rate of change is represented by the steepest of the red line segments. Also notice that between 1988 and 1990, when the cost did not change, the red line segment is horizontal. CONFIDENTIAL

  30. Slope of a Line If all of the segments have the same rate of change, then they all have the same steepness and together form a straight line. The constant rate of change of a line is called the slope of the line. The rise is the difference in the y-values of two points on a line. The run is the difference in the x-values of two points on a line. The slope of a line is the ratio of rise to run for any two points on the line. slope = rise = Change in y run Change in x CONFIDENTIAL Next slide 

  31. (Remember that yis the dependent variableand xis the independent variable.) CONFIDENTIAL

  32. Finding Slope 3) Find the slope of the line. Begin at one point and count vertically to find the rise. Then count horizontally to the second point to find the run. It does not matter which point you start with. The slope is the same. slope = 2 = 2 1 slope = -2 = 2 -1 CONFIDENTIAL

  33. As shown in the previous examples, slope can be positive, negative, zero, or undefined. You can tell which of these is the case by looking at the graph of a line—you do not need to calculate the slope. CONFIDENTIAL

  34. Describing Slope 5) Tell whether the slope of each line is positive, negative, zero, or undefined. A) B) The line falls from left to right. The slope is negative. The line is horizontal. The slope is 0. CONFIDENTIAL

  35. The line with slope 4 is steeper than the line with slope 1 | 4 | > 1 2 2 Comparing Slope Remember that the slope of a line is its steepness. Some lines are steeper than others. As the absolute value of the slope increases, the line becomes steeper. As the absolute value of the slope decreases, the line becomes less steep. CONFIDENTIAL

  36. You did a great job today! CONFIDENTIAL