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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 the following: • Function: • Linear function: • Direct Variation:

  9. $100 Answer Linear Functions • Function: passes VLT • Linear function: passes VLT & straight line • Direct Variation: passes VLT, straight line & crosses through (0,0)

  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 • Table A – not a constant rate of change

  12. $300 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

  13. $300 Answer Linear Functions • y = 2x2 – 7not 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

  14. $400 Question Linear Functions Explain if each equation represents a direct variation (proportional relationship). Identify the constant of variation if it is a direct variation. A.) 3y = 4x + 1 B.) y + 3x = 0

  15. $400 Answer Linear Functions Solve each equation for y. Look to see if it’s in y = kx form k = constant of variation B.) y + 3x = 0 y = -3x Yes, in y = kx form k = m -3 A.) 3y = 4x + 1 y = 4/3x + 1 Not in y = kx form (+1)

  16. $500 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

  17. $500 Answer Linear Functions • As x goes up by 6 • The value of y. c.) increases by 12

  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 • (6, 5) and (1, 2)

  21. $200 Answer Slope 1. Label Points 2. Use slope formula Remember neg. divided by a neg. is positive

  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 = 10h + 100, • where h 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

  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 • 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 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 dependent variable (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 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 = [x – (-3)] y – 4 = (x + 3) Slope-Intercept Form y = x + 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 Solving Equations and Inequalities Review Solve and Graph -2m < 8

  49. $100 Answer Solving Equations and Inequalities Review -2m < 8 - 2 -2 m > -4 Open to the right

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