Linear Functions - ACT Math

1. What is the point-slope form of a linear equation through $(x_1,y_1)$ with slope $m$?

   $y-y_1=m(x-x_1)$. Uses known point $(x_1,y_1)$ and slope $m$.

2. What is the equation of the line with $x$-intercept $-2$ and $y$-intercept $6$?

   $y=3x+6$. Slope is $\frac{6-0}{0-(-2)}=3$.

3. What is the $y$-intercept of the line $y=mx+b$?

   $b$. The constant term when $x=0$.

4. What is the $x$-intercept of the line $y=-3x+6$?

   $2$. Set $y=0$: $0=-3x+6$, so $x=2$.

5. What is the equation of the line through points $(2,1)$ and $(2,5)$?

   $x=2$. Same $x$-coordinate means vertical line.

6. Find the y-intercept of the line $5x + 2y = 10$.

   5. Set $x = 0$: $2y = 10$, so $y = 5$.

7. What is the $y$-intercept of the line through $(2,3)$ with slope $4$?

   $-5$. Use point-slope: $3=4(2)+b$, so $b=-5$.

8. Is the line $y = \frac{3}{4}x - 2$ increasing or decreasing?

   Increasing. Positive slope means line goes up left to right.

9. What is the $x$-intercept of $Ax+By=C$ (with $A\ne 0$)?

   $\frac{C}{A}$. Set $y=0$ in standard form and solve.

10. What is the average rate of change of $f(x)=-3x+2$ from $x=1$ to $x=4$?

    $-3$. For linear functions, rate of change equals slope.

11. Determine if $y = \frac{3}{2}x + 5$ and $y = -\frac{2}{3}x + 1$ are perpendicular.

    Yes. Slopes $\frac{3}{2}$ and $-\frac{2}{3}$ multiply to $-1$.

12. What is the slope of a line through $(2, 3)$ and $(4, 3)$?

    0. Same $y$-coordinate means horizontal line.

13. What is the $x$-intercept of the line $y=mx+b$ in terms of $m$ and $b$ (assuming $m\ne 0$)?

    $x=-\frac{b}{m}$. Set $y=0$ and solve for $x$.

14. Convert $y = 2x + 3$ to standard form.

    $-2x + y = 3$. Move $x$ term to left side.

15. What is the slope-intercept form of a linear equation?

    $y = mx + b$. Form showing slope $m$ and y-intercept $b$ directly.

16. What is the slope of a horizontal line?

    $0$. No vertical change means zero slope.

17. Find the slope of the line passing through $ (0, 0) $ and $ (5, 10) $.

    2. Using slope formula: $\frac{10-0}{5-0} = 2$.

18. What is the equation of a horizontal line passing through $y=k$?

    $y=k$. Constant $y$-value for all $x$.

19. What is the $y$-intercept of the line $y=\frac{1}{4}x-8$?

    $-8$. The constant term in slope-intercept form.

20. What is the equation of the line passing through $(-3,2)$ with undefined slope?

    $x=-3$. Undefined slope means vertical line.

21. What is the slope of a vertical line?

    Undefined. Division by zero in slope formula.

22. Find the x-intercept of $4x - y = 8$.

    2. Set $y = 0$: $4x = 8$, so $x = 2$.

23. What condition on slopes $m_1$ and $m_2$ means two nonvertical lines are perpendicular?

    $m_1m_2=-1$. Perpendicular slopes are negative reciprocals.

24. What is the slope of a line parallel to the line $y = \frac{5}{2}x + 1$?

    $\frac{5}{2}$. Parallel lines have identical slopes.

25. State the formula to calculate the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$.

    $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run between any two points.

26. What is $x$ if the point $(x,9)$ lies on the line $y=2x+1$?

    $4$. Solve: $9=2x+1$, so $x=4$.

27. Find the slope of the line $x = 5$.

    Undefined. Vertical line has undefined slope.

28. What is the x-intercept of the equation $y = 3x + 9$?

    -3. Set $y = 0$: $0 = 3x + 9$, so $x = -3$.

29. Determine if the lines $y = 2x + 3$ and $y = 2x - 4$ are parallel.

    Yes. Both have slope $2$, so they're parallel.

30. Convert $3x + 4y = 12$ to slope-intercept form.

    $y = -\frac{3}{4}x + 3$. Solve for $y$: $4y = -3x + 12$.

31. What is the equation of the line with slope $-4$ passing through $(0,7)$?

    $y=-4x+7$. Point $(0,7)$ gives $y$-intercept directly.

32. What is the slope of the line passing through $(2,5)$ and $(6,13)$?

    $2$. Using slope formula: $\frac{13-5}{6-2}=\frac{8}{4}=2$.

33. What is the slope of the line $y=-7$?

    $0$. Horizontal line has zero slope.

34. Identify the slope of the line $3y - 9x = 12$.

    3. Rewrite as $y = 3x + 4$; slope is $3$.

35. What is the equation in slope-intercept form of the line with slope $3$ and $y$-intercept $-2$?

    $y=3x-2$. Direct substitution into $y=mx+b$.

36. What is the slope of the line through $(0,-1)$ and $(8,3)$?

    $\frac{1}{2}$. Using slope formula: $\frac{3-(-1)}{8-0}=\frac{4}{8}=\frac{1}{2}$.

37. What is the average rate of change of a linear function $f$ on $[a,b]$?

    $\frac{f(b)-f(a)}{b-a}$. For linear functions, this equals the slope.

38. What is the slope-intercept form of the line $2x+3y=12$?

    $y=-\frac{2}{3}x+4$. Divide by $3$: $y=-\frac{2}{3}x+4$.

39. What is the slope of the line given by the equation $4x - 2y = 8$?

    2. Rewrite as $y = 2x - 4$; slope is $2$.

40. Find the slope of a vertical line.

    Undefined. Division by zero in slope formula.

41. What is the equation of the line through $(2, -1)$ with slope $\frac{1}{2}$?

    $y + 1 = \frac{1}{2}(x - 2)$. Point-slope form with given point and slope.

42. What is the slope of the line passing through $(-1,4)$ and $(3,0)$?

    $-1$. Using slope formula: $\frac{0-4}{3-(-1)}=\frac{-4}{4}=-1$.

43. Determine if $y = 4x + 1$ and $y = 4x - 3$ are parallel.

    Yes. Both have slope $4$, so they're parallel.

44. What is the slope of the line $y = 7$?

    0. Horizontal line has zero slope.

45. What is the equation of a line passing through $(0, -2)$ with slope $-3$?

    $y = -3x - 2$. Line passes through $(0, -2)$ with given slope.

46. Convert the equation $2x + 3y = 6$ to slope-intercept form.

    $y = -\frac{2}{3}x + 2$. Solve for $y$: $3y = -2x + 6$, so $y = -\frac{2}{3}x + 2$.

47. Is the line $y = -\frac{1}{2}x + 3$ increasing or decreasing?

    Decreasing. Negative slope means line goes down left to right.

48. What is the $y$-intercept of $Ax+By=C$ (with $B \neq 0$)?

    $\frac{C}{B}$. Set $x=0$ in standard form and solve.

49. What is the equation of the line with slope $2$ passing through $(3,-1)$ in point-slope form?

    $y+1=2(x-3)$. Direct substitution into point-slope form.

50. What is the standard form of a linear equation?

    $Ax + By = C$. General form with integer coefficients $A$, $B$, and $C$.

51. What is the $y$-intercept of the line $4x-2y=10$?

    $-5$. Convert to slope-intercept: $y=2x-5$.

52. What does the variable $b$ represent in the equation $y = mx + b$?

    The y-intercept of the line. Constant term where line crosses y-axis at $(0, b)$.

53. What is the equation of the line with slope $-1$ and $x$-intercept $4$?

    $y=-x+4$. Line passes through $(4,0)$ and $(0,4)$.

54. What is the equation of a vertical line passing through $x=h$?

    $x=h$. Constant $x$-value for all $y$.

55. What is the equation of the line through $(2,3)$ with slope $4$ in slope-intercept form?

    $y=4x-5$. From point-slope: $y-3=4(x-2)$.

56. What is the slope of a line perpendicular to $y=\frac{2}{5}x+1$?

    $-\frac{5}{2}$. Negative reciprocal of $\frac{2}{5}$.

57. What condition on slopes $m_1$ and $m_2$ means two nonvertical lines are parallel?

    $m_1=m_2$. Parallel lines have identical slopes.

58. What does the variable $m$ represent in the equation $y = mx + b$?

    The slope of the line. Coefficient of $x$ determines steepness and direction.

59. Find the slope between points $(2, 3)$ and $(4, 7)$.

    2. Using slope formula: $\frac{7-3}{4-2} = \frac{4}{2} = 2$.

60. Find the x-intercept of the line $3x - y = 9$.

    3. Set $y = 0$: $3x = 9$, so $x = 3$.

61. What is the equation of a line with slope 2 passing through $(1, 3)$?

    $y - 3 = 2(x - 1)$. Point-slope form: $y - y_1 = m(x - x_1)$.

62. What is the slope of the line through $(0,b)$ and $(1,m+b)$?

    $m$. Using slope formula: $\frac{(m+b)-b}{1-0}=m$.

63. What is the equation of the line through points $(1,2)$ and $(3,6)$ in slope-intercept form?

    $y=2x$. Slope is $2$, passes through origin.

64. What is the slope of a line that passes through $(1, 2)$ and $(1, 5)$?

    Undefined. Same $x$-coordinate means vertical line.

65. What is the slope of the line $x=9$?

    Undefined. Vertical line has undefined slope.

66. Determine if the lines $y = -\frac{1}{2}x + 1$ and $y = 2x - 3$ are perpendicular.

    Yes. Slopes are $-\frac{1}{2}$ and $2$; their product is $-1$.

67. What is the slope of the line segment from $(4,-2)$ to $(0,6)$?

    $-2$. Using slope formula: $\frac{6-(-2)}{0-4}=\frac{8}{-4}=-2$.

68. What is $f(5)$ if $f(x)=2x-3$?

    $7$. Substitute: $f(5)=2(5)-3=7$.

69. Find the slope of the line passing through $(0, 0)$ and $(5, 10)$.

    2. Using slope formula: $\frac{10-0}{5-0} = 2$.

70. What is the slope of a line perpendicular to the line $y = -3x + 4$?

    $\frac{1}{3}$. Negative reciprocal of $-3$ is $\frac{1}{3}$.

71. Determine if the lines $2x + 3y = 6$ and $4x + 6y = 12$ are identical.

    Yes. Second equation is twice the first; same line.

72. Convert the equation $y = -2x + 5$ to standard form.

    $2x + y = 5$. Move $x$ term: $2x + y = 5$.

73. Find the slope of a horizontal line.

    0. No vertical change means slope equals zero.

74. What is the slope of the line through $ (0,b) $ and $ (1,m+b) $?

    $m$. Using slope formula: $\frac{(m+b)-b}{1-0}=m$.

75. What is the equation of the line passing through $(1,5)$ with slope $0$?

    $y=5$. Zero slope means horizontal line.

76. What is the slope of the line $5x+10y=20$?

    $-\frac{1}{2}$. Convert to slope-intercept: $y=-\frac{1}{2}x+2$.

77. Identify the y-intercept in the equation $y = 4x - 7$.

    -7. Constant term in slope-intercept form.

78. Identify the slope in the equation $y = -3x + 5$.

    -3. Coefficient of $x$ in slope-intercept form.

79. What is the equation of the line through points $(-3,4)$ and $(1,4)$?

    $y=4$. Same $y$-coordinate means horizontal line.

80. What is the slope of the line represented by $Ax+By=C$ (with $B\ne 0$)?

    $m=-\frac{A}{B}$. Rearrange to slope-intercept form.

81. What is the slope of a line parallel to $3x-6y=12$?

    $\frac{1}{2}$. Parallel lines have the same slope.

82. What is the equation of the line through $(0, 0)$ and slope $-4$?

    $y = -4x$. Line through origin with slope $-4$.

83. What is the y-intercept of the equation $4x + 2y = 8$?

    4. Set $x = 0$: $2y = 8$, so $y = 4$.

84. What is the slope formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points.

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