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  1. Slope & y-intercept on the Coordinate Plane Return to Table of Contents Day 2

  2. Consider this graph of the Cartesian Plane, also called a Coordinate Plane or XY-Plane. Imagine trying to tell a person how to draw a line on the Cartesian Plane.

  3. The Equation of a Line y You only need a few facts about a line to completely describe it: ·Its y-intercept (where it crosses the y-axis) "b" · Its slope (how much it rises or falls) "m" · y = mx + b 10 8 6 4 2 x 0 -8 -4 -2 -10 -6 2 4 6 10 8 -2 -4 -6 -8 -10

  4. The y-intercept y The y-intercept ("b")of a line is the point where the line intercepts the y-axis. In this case, the y-intercept of the line is +4. 10 8 6 4 This is the ordered pair (0,4). 2 x 0 -8 -4 -2 -10 -6 2 4 6 10 8 -2 -4 -6 -8 -10

  5. 11 What is the y-intercept of this line? b = 6 Answer (click me)

  6. 12 What is the y-intercept of this line? b = -4 Answer (click me)

  7. 13 What is the y-intercept of this line? b = 8 Answer (click me)

  8. y 14 What is the x-intercept of this line? 10 8 6 4 2 x 0 -8 -4 -2 -10 -6 2 4 6 10 8 -2 -4 (-4,0) -6 Answer (click me) -8 -10

  9. 15 What is the x-intercept of this line? (3,0) Answer (click me)

  10. 16 What is the x-intercept of this line? (-6,0) Answer (click me)

  11. 17 The graph of the equation x + 3y = 6 intersects the y-axis at the point whose coordinates are A (0,2) B (0,6) C (0,18) D (6,0) From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

  12. Defining Slope on the Coordinate Plane Return to Table of Contents

  13. "Steepness" and "Position" of a Line

  14. Consider this... y An infinite number of lines can pass through the same location on the y-axis...they all have the same y-intercept. Examples of lines with a y-intercept of ____ are shown on this graph. What's the difference between them (other than their color)? 10 8 6 4 2 x 0 -8 -4 -2 -10 -6 2 4 6 10 8 -2 -4 -6 -8 -10

  15. The Slope of a Line y The lines all have a different slope. Slope is the steepness of a line. Compare the steepness of the lines on the right. Slope can also be thought of as the rate of change. 10 8 6 4 2 x 0 -8 -4 -2 -10 -6 2 4 6 10 8 -2 -4 -6 -8 -10

  16. The Slope of a Line run y The red line has a positive slope, since the line rises from left to the right. 10 rise 8 6 4 2 x 0 -8 -4 -2 -10 -6 2 4 6 10 8 -2 -4 -6 -8 -10

  17. The Slope of a Line y The orange line has a negative slope, since the line falls down from left to the right. 10 8 6 rise 4 run 2 x 0 -8 -4 -2 -10 -6 2 4 6 10 8 -2 -4 -6 -8 -10

  18. The Slope of a Line y 10 The purple line has a slope of zero, since it doesn't rise at all as you go from left to right on the x-axis. 8 6 4 2 x 0 -2 -8 -4 -10 -6 2 4 6 10 8 -2 -4 -6 -8 -10

  19. The Slope of a Line The black line is a vertical line. It has an undefined slope, since it doesn't run at all as you go from the bottom to the top on the y-axis. rise 0 = undefined

  20. 18 Is the slope of the following graph positive, negative, zero, or undefined? A positive B negative C zero D undefined

  21. 19 Is the slope of the following graph positive, negative, zero, or undefined? A positive B negative C zero D undefined

  22. 20 Is the slope of the following graph positive, negative, zero, or undefined? A positive B negative C zero D undefined

  23. 21 Is the slope of the following graph positive, negative, zero, or undefined? A positive B negative C zero D undefined

  24. 22 Is the slope of the following graph positive, negative, zero,or undefined A positive B negative C zero D undefined

  25. Is the slope of the following graph positive, negative, zero, or undefined? 23 A positive B negative C zero D undefined

  26. 24 Is the slope of the following graph positive, negative, zero, or undefined? A positive B negative C zero D undefined

  27. 25 Is the slope of the following graph positive, negative, zero, or undefined? A positive B negative C zero D undefined

  28. Measuring the Slope of a Line While we can quickly see if the slope of a line is positive, negative or zero...we also need to determine how much slope it has...we have to measure the slope of a line.

  29. Measuring the Slope of a Line The slope of the line is just the ratio of its rise over its run. The symbol for slope is "m". So the formula for slope is: rise slope = run

  30. Measuring the Slope of a Line rise slope = run The slope is the same anywhere on a line, so it can be measured anywhere on the line. Keep in mind the direction: · Up (+) Down (-) · Right (+) Left (-)

  31. Measuring the Slope of a Line For instance, in this case we measure the slope by using a run from x = 0 to x = +6: a run of 6. During that run, the line rises from y = 0 to y = 8: a rise of 8. rise 8 slope = m = run 6 4 m = 3

  32. Measuring the Slope of a Line But we get the same result with a run from x = 0 to x = +3: a run of 3. During that run, the line rises from y = 0 to y = 4: a rise of 4. rise 4 m = slope = 3 run

  33. Measuring the Slope of a Line But we can also start at x = 3 and run to x = 6 : a run of 3. During that run, the line rises from y = 3 to y = 7: a rise of 4. 4 m = 3 rise slope = run

  34. Measuring the Slope of a Line But we can also start at x = -6 and run to x = 0: a run of 6. During that run, the line rises from y = -8 to y = 0: a rise of 8. 8 m = 6 rise slope = 4 run m = 3

  35. Measuring the Slope of a Line How is the slope different on this coordinate plane? The line rises 8, however the run goes left 6 (negative). Therefore, it is said to have a negative slope rise slope = run 8 m = -6 -4 m = 3 *most often the negative sign is placed in the numerator

  36. 26 What's the slope of this line?

  37. 27 What's the slope of this line?

  38. 28 What's the slope of this line?

  39. 29 What's the slope of this line?

  40. 30 What's the slope of this line?

  41. 31 What's the slope of this line? A -20 B 0 C 20 D undefined

  42. 32 In the diagram below, what is the slope of the line passing through points A and B? A -2 B . B 2 A C -1/2 D 1/2 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

  43. 33 A straight line with slope 5 contains the points (1,2) and (3,K). Find the value of K. [The use of the accompanying grid is optional.] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

  44. Tables and Slope Return to Table of Contents

  45. How can slope and the y-intercept be found within the table? · Look for the change in the y-values · Look for the change in the x-values · Write as a ratio (simpified) - this will be the "slope" · Determine the corresponding y-value to the x-value of 0 - this will be the "y-intercept"

  46. +3 +6 +3 +6 5 is the y-intercept 6 3 2 is the slope =

  47. Determine the slope and y-intercept from this table. is the slope -4 is the y-intercept click to reveal answer

  48. 34 What is the slope?

  49. 35 What is the y-intercept? You may have to do a little extra work for this one...

  50. 36 What is the slope?