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Linear Functions

Linear Functions. Slope. Parallel & Perpendicular Lines. Different Forms Of Linear Equations. Graphing Linear Inequalities Systems of Equations. Miscellaneous . Linear Functions. Parallel & Perpendicular Lines. Different Forms of Linear Equations. Graphing

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Linear Functions

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  1. Linear Functions

  2. Slope

  3. Parallel & Perpendicular Lines

  4. Different Forms Of Linear Equations

  5. Graphing Linear Inequalities Systems of Equations

  6. Miscellaneous

  7. Linear Functions Parallel & Perpendicular Lines Different Forms of Linear Equations Graphing Linear Inequalities and Systems Misc. Linear Functions Slope $100 $100 $100 $100 $100 $100 $200 $200 $200 $200 $200 $200 $300 $300 $300 $300 $300 $300 $400 $400 $400 $400 $400 $400 $500 $500 $500 $500 $500 $500

  8. $100 Question Linear Functions Explain how you know if a graph represents a linear function

  9. $100 Answer Linear Functions A graph represents a linear function if it is a straight line **vertical lines are linear but not functions (fails vertical line test.)

  10. $200 Question Linear Functions Which table(s) are linear? Explain how you know. B. A.

  11. $200 Answer Linear Functions B.) x and y are both going up with constant rates * same rate of change * Rate of change = change of y change of x x is not a constant change in table A

  12. $300 Question Linear Functions Make a table and graph for y = -x + 3 Is this equation linear? Explain.

  13. $300 Answer Linear Functions Graph has a negative slope—straight line It is a linear function because table has a constant rate of change and graph is a straight line

  14. $400 Question Linear Functions Let y = 2x + 9. If the value of x increases by 6, which of the following best describes the change in the value of y. a.) decreases by 6 b.) increases by 6 c.) increases by 12 d.) increases by 21

  15. $400 Answer Linear Functions As x goes up by 6 The value of y. c.) increases by 12

  16. $500 Question Linear Functions • Which of the following equations is not linear? • Explain or show how you know. • y = 2x2 – 7 • -6= y • 4x – 2y = 10 • y = 3x + 1

  17. $500 Answer Linear Functions • y = 2x2 – 7 not linear--exponent • -6= y horizontal—straight line • 4x – 2y = 10standard form x and y-int. --line • y = 3x + 1 slope-int.—always form a line

  18. $100 Question Slope Find the slope of the line (0, 4) (3, -5)

  19. $100 Answer Slope Rise m = -3 Run

  20. $200 Question Slope Write the equation of the line that passes through each pair of points in slope-intercept form (-1, 5) and (2, -4)

  21. $200 Answer Slope 1.) Find Slope m = -3 2. ) Choose a point (-1,5) Use point-slope form then solve for y Y = -3x + 2

  22. $300 Question Slope Put the following equation into slope-intercept form. Identify the slope and y-intercept. Then use the slope and y-int. to graph the line. 3x – y = 2

  23. $300 Answer Slope 3x – y = 2 y = 3x -2 m = 3 -3x -3x b = -2 -y = -3x + 2 -1 -1 -1

  24. $400 Question Slope Laurel graphed the equation y = -2x + 5. Katelyn then graphed an equation that was a line that was not as steep as Laurel’s. Which equation could have been the one Katelyn graphed? a.) y = -3x + 5 b.) y = 1/2x + 6 c.) y = 4x – 2 d.) y = -2x + 3

  25. $400 Answer Slope B.) y = 1/2x + 6 is not as steep. Fractions (between -1 and 1: non-improper) are less steep than any integer— even if it’s negative.

  26. $500 Question Slope The cost of hiring Zach as a painter is given by the linear equation C = 10t + 100, where t is the number of hours Zach works. Identify the slope and y-int. What does the slope of the line represent? What does the y-intercept represent?

  27. $500 Answer Slope m = 10 The slope means Zach earns $10 per hour. b = 100 The y-intercept represents base charge of hiring Zach (when he’s worked 0 hours, we’d still have to pay him $100

  28. $100 Question Parallel & Perpendicular Lines What are two different ways that lines can be perpendicular?

  29. $100 Answer Parallel & Perpendicular Lines Vertical lines are perpendicular to a horizontal lines. Ex. x = 3 and y = -2 When the product of slopes = -1 (or are negative reciprocals of each other) Ex. 4 and -1/4

  30. $200 Question Parallel & Perpendicular Lines A line has the equation x + 2y = 5 What is the slope of a line parallel to this line? a.) – 2 b.) - ½ c.) ½ d.) 2

  31. $200 Answer Parallel & Perpendicular Lines • A line has the equation x + 2y = 5 • Put line in slope-int. form y = -1x + 5 • 2 2 • 2. Parallel -- same slope -- b.) - ½

  32. $300 Question Parallel & Perpendicular Lines

  33. $300 Answer Parallel & Perpendicular Lines A. 1 and -1 are “opposite reciprocals”

  34. $400 Question Parallel & Perpendicular Lines

  35. $400 Answer Parallel & Perpendicular Lines Line AB has a slope of 1 and Line BC has a slope of -3/2 and Line AC has a slope of 0. None of the slopes will have a product of -1 (are negative reciprocals) so D is the answer

  36. $500 Question Parallel & Perpendicular Lines Write an equation that is perpendicular to the given line below that passes through the point (- 6, 2)

  37. $500 Answer Parallel & Perpendicular Lines • Slope will be -3 (opp. reciprocal) • Use point-slope form • y- 2 = -3(x – (-6)) Distributive Prop. • y = -3x -16

  38. $100 Question Different Forms of Linear Equations Find and use the x and y intercepts to graph the line. -x + 3y = 6

  39. $100 Answer Different Forms of Linear Equations -x + 3y = 6 -x + 3y = 6 0 + 3y = 6 -x + 3(0) = 6 y-int. = 2 -x = 6 (0,2) x-int. = -6 (-6,0) (0,2) (-6,0)

  40. $200 Question Different Forms of Linear Equations Find and use the x and y intercepts to graph the line. -2x = 12 + 4y

  41. $200 Answer Different Forms of Linear Equations -2x = 12 + 4y -4y from both sides -2x - 4y = 12 Now in standard form x-int. = (-6,0) y-int. = (0,-3) (-6,0) (0, -3)

  42. $300 Question Different Forms of Linear Equations What is the x-intercept of the linear function f(x) = -3x + 6? Note: f(x) is another way to write ‘y’ a.) -2 b.) 2 c.) 3 d.) 6

  43. $300 Answer Different Forms of Linear Equations f(x) = -3x + 6 Think: y = -3x + 6 add 3x to both sides –standard form y + 3x = 6 0 + 3x = 6 x-int. = 2 (b)

  44. $400 Question Different Forms of Linear Equations A line has a slope of 2/3 and passes through the point (-3, 4). What is the equation of the line in point-slope form? What is the equation of the line in slope-intercept form?

  45. $400 Answer Different Forms of Linear Equations Point-slope form y- 4 = 2/3[x – (-3)] y – 4 = 2/3(x + 3) Slope-Intercept Form y = 2/3x + 6

  46. $500 Question Different Forms of Linear Equations Is every linear relationship a direct variation? Is every direct variation a linear relationship? Explain.

  47. $500 Answer Different Forms of Linear Equations Every linear relationship is not a direct variation—only if the y-int. is 0. However, every direct variation is linear because it has a constant rate of change. Direct Variation: y = 3x (also linear) Not a direct variation y = 3x +5 (is linear)

  48. $100 Question Graphing Linear Inequalities and Systems of Equations Graph each inequality y > -3x + 2

  49. $100 Answer Graphing Linear Inequalities and Systems of Equations • m = -3 b = 2 • Dashed • (0,0) not a solution

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