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Chapter 6

Chapter 6. Linear Functions. 6.1 – slope of a line. Chapter 6. Slope – steepness. Some roofs are steeper than others. Steeper roofs are more expensive to shingle. So, what are the slopes for these three roofs?. example. Determine the slope of each line segment. a). b). Try it.

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Chapter 6

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  1. Chapter 6 Linear Functions

  2. 6.1 – slope of a line Chapter 6

  3. Slope – steepness Some roofs are steeper than others. Steeper roofs are more expensive to shingle. So,what are the slopes for these three roofs?

  4. example Determine the slope of each line segment. a) b)

  5. Try it Determine the slope of each line segment. a) b)

  6. Horizontal/vertical lines For a horizontal line segment:  What’s the rise?  What’s the run? • For a vertical line segment: • What’s the rise? • What’s the run? Can you divide by zero?

  7. example Draw a line segment with each given slope. a) b) rise rise a) b) run run

  8. P. 339-340, #5, 6, 7, 9, 11 Independent practice

  9. example Determine the slope of the line that passes through C(–5, –3) and D(2, 1).

  10. formula I would encourage everyone to solve for slope using the simple “rise over run” formula, and drawing a picture when necessary. However, there is a formula for those of you who like them. A line passes through A(x1,y1) and B(x2,y2).

  11. Try it Determine the slope of the line that passes through E(4, –5) and F(8, 6).

  12. example Yvonne recorded the distances she had travelled at certain times since she began her cycling trip along the Trans Canada Trail in Manitoba, from North Winnipeg to Grand Beach. She plotted these data on a grid. What is the slope of the line through these points? What does the slope represent? How can the answer to part b be used to determine:i) how far Yvonne travelled in 1.75 hours?ii) the time it took Yvonne to travel 55km?

  13. Pg. 340-343, #13, 16, 17, 18, 20, 25, 30. Independent practice

  14. 6.2 – slopes of parallel and perpendicular lines Chapter 6

  15. Parallel lines What does parallel mean? Parallel lines are two lines on a plane that never meet, and are always the same distance apart. What is the slope of these two lines?

  16. example A line GH passes through G(–4, 2) and H(2, –1). Line JK passes through J(–1, 7) and K(7,3). Line MN passes through M(–4, 5) and N(5, 1). Sketch the lines. Are they parallel? Justify your answer. To show that lines are parallel, you need to show that they have the same slopes. Line EF passes through E(–3, –2) and F(–1, 6). Line CD passes through C(–1, –3) and D(1, 7). Line AB passes through A(–3, 7) and B(–5, –2). Sketch the lines. Are they parallel?

  17. challenge Are these lines parallel? Show work.

  18. Perpendicular lines Non-parallel lines in the same plane have different slopes. Perpendicular lines are not parallel, so they have different slopes. What does perpendicular mean? Perpendicular lines are lines that form a 90º angle where they meet. The slopes of perpendicular lines are negative reciprocals of one another.

  19. example Line PQ passes through P(–7, 2) and Q(–2, 10). Line RS passes through R(–3, –4) and S(5, 1). Are these two lines parallel, perpendicular, or neither? Justify. Sketch the lines to verify the answer to part A. Line ST passes through S(–2, 7) and T(2, –5). Line UV passes through U(–2, 3) and V(7, 6). Are these lines parallel, perpendicular, or neither?

  20. example Determine the slope of a line that is perpendicular to the line through E(2, 3) and F(–4, –1). Determine the coordinates of G so that line EG is perpendicular to line EF. What is the slope of EF? Use the formula: Draw it out: What’s the negative reciprocal of it’s slope?

  21. example ABCD is a parallelogram. Is it a rectangle? Justify your answer.

  22. Pg. 349-351, #5, 6, 8, 9, 12, 13, 17, 19. Independent practice

  23. 6.3 – investigating graphs of linear functions Chapter 6

  24. challenge ABCD is a parallelogram. Three vertices have coordinates A(–4, 3), B(2, 4), and C(4, 0). Is ABCD a rectangle? Justify your answer. Determine the coordinates of D.

  25. Linear functions Alimina purchased an mp3 player and downloaded 3 songs. Each subsequent day, she downloads 2 song. Which graph represents this situation? Explain your choice.

  26. Pg. 356, #3-6 Independent practice

  27. 6.4 – slope-intercept form of the equation for a linear function Chapter 6

  28. Linear functions This graph shows a cyclist’s journey where the distance is measured from her home. What does the vertical intercepts represent? What does the slope of the line represent?

  29. Slope-intercept form The equation of a linear function can be written in the form y = mx + b, where m is the slope of the line and b is its y-intercept.

  30. example The graph of a linear function has slope and y-intercept –4. Write an equation. The graph of a linear function has slope and y-intercept 5. Write an equation.

  31. Write the equation for this graph.

  32. example Graph the linear function with equation: rise run slope = ½ y-intercept = 3 Graph:

  33. example Write and equation to describe this function. Verify by checking a point on the graph.

  34. example The student council put on a dance. A ticket cost $5 and the cost for the DJ was $300. Write an equation for the profit, P dollars, on the sale of t tickets. Suppose 123 people bought tickets. What was the profit? Suppose the profit was $350. How many people bought tickets? Could the profit be exactly $146? Justify the answer.

  35. Pg. 362-364, #4, 5, 7, 8, 12, 13, 16, 18, 19. Independent practice

  36. 6.5 – slope-point form of the equation for a linear function Chapter 6

  37. Point-slope form The line has slope –3 and passes through P(–2, 5). We use any other point Q(x, y) on the line to write an equation for the slope, m: This form is called slope-point form. It’s used when you have the slope and a point, but not the y-intercept.

  38. Slope-point form The equation of a line that passes through P(x1,y1) and has slope mis:

  39. example Describe the graph of the linear function with this equation: Graph the equation. b) a) Remember that the x-value of the point is always the opposite sign. What’s it slope? What point does it go through?

  40. example Write an equation in slope-point form for this line. Write the equation in part a in slope-intercept form. What is the y-intercept of this line? What’s the slope? What’s the point (x1,y1)? We can’t see the y-intercept, so we need to use point-slope form.

  41. example Write an equation in slope-point form for this line. Write the equation in part a in slope-intercept form. What is the y-intercept of this line? b)

  42. example Write an equation for the line that passes through R(1, –1) and is: Parallel to the line Perpendicular to the line

  43. Pg. 372-374, #4, 7, 9, 11, 14, 17, 20, 23 Independent practice

  44. challenge Find the equation of the line with a slope of –1/3 going through point (–4,7)

  45. 6.6 – general form of the equation for a linear relation Chapter 6

  46. General form Ax + By + C = 0 is the general form of the equation of a line, where A is a whole number, and B and Care integers.

  47. example Try these: Write each equation in general form. a) b) a) Firstly, we want whole numbers, so we need to multiply by the denominator. b) Multiply by the denominator.

  48. challenge Write the following equation in both slope-intercept form and in general form:

  49. example Determine the x- and y-intercepts of the line whose equation is: 3x + 2y – 18 = 0 Graph the line. Verify that the graph is correct. To determine the x-intercept, let y = 0 3x + 2y – 18 = 0 3x + 2(0) – 18 = 0 3x – 18 = 0 3x = 18 x = 6 To find the y-intercept, let x = 0 3x + 2y – 18 = 0 3(0) + 2y – 18 = 0 2y – 18 = 0 2y = 18 y = 9

  50. example Determine the x- and y-intercepts of the line whose equation is: 3x + 2y – 18 = 0 Graph the line. Verify that the graph is correct. We found that: x = 6y = 9

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