Graphs - PSAT Math

1. What is the distance between $(1,-2)$ and $(4,2)$?

   $5$. Apply distance formula: $\sqrt{(4-1)^2+(2-(-2))^2}=\sqrt{9+16}$.

2. What is the vertex form of a quadratic function?

   $y=a(x-h)^2+k$. Shows parabola with vertex at $(h,k)$ and vertical stretch $a$.

3. Identify the transformation of $y=-f(x)$ relative to $y=f(x)$.

   Reflect across the $x$-axis. Negative sign flips all $y$-values across horizontal axis.

4. What is the equation of a line with slope $3$ and $y$-intercept $-4$?

   $y=3x-4$. Substitute $m$ and $b$ into $y=mx+b$.

5. What is the slope of a line parallel to $y=-\frac{1}{3}x+7$?

   $m=-\frac{1}{3}$. Parallel lines have identical slopes.

6. State the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$.

   $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Apply the Pythagorean theorem to find diagonal distance.

7. What is the slope of a vertical line (for example, $x=-4$)?

   Slope is undefined. Vertical lines have infinite rise over zero run.

8. What is the slope of a horizontal line (for example, $y=7$)?

   $m=0$. Horizontal lines have no rise, so slope equals zero.

9. What is the slope of a line passing through $(2,5)$ and $(6,1)$?

   $-1$. Using slope formula: $\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

10. What is the equation of the vertical line passing through $(4,0)$?

    $x=4$. All vertical lines have form $x=k$ for constant $k$.

11. Identify the slope of the line through $(2,3)$ and $(6,11)$.

    $m=2$. $m=\frac{11-3}{6-2}=\frac{8}{4}=2$.

12. What is the average rate of change of $f(x)=x^2$ from $x=1$ to $x=3$?

    $4$. $\frac{f(3)-f(1)}{3-1}=\frac{9-1}{2}=4$.

13. State the point-slope form of a line with slope $m$ through $(x_1,y_1)$.

    $y-y_1=m(x-x_1)$. Expresses a line using a known point and the slope.

14. Find the distance between $(1,2)$ and $(4,6)$.

    $5$. $d=\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=\sqrt{25}=5$.

15. Identify the $x$-intercept of the line $y=-2x+8$.

    $(4,0)$. Set $y=0$: $0=-2x+8$, so $x=4$.

16. What is the midpoint formula for the segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

    $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the x-coordinates and y-coordinates separately.

17. What is the vertex of $y=(x-4)^2+1$?

    $(4,1)$. Vertex form $(x-h)^2+k$ has vertex at $(h,k)$.

18. What is the slope of a line parallel to $y=-3x+7$?

    $-3$. Parallel lines have identical slopes.

19. Find the midpoint of the segment with endpoints $(2,-1)$ and $(8,5)$.

    $(5,2)$. $\left(\frac{2+8}{2},\frac{-1+5}{2}\right)=(5,2)$.

20. Find the slope of the line through $(2,5)$ and $(6,1)$.

    $m=-1$. Using slope formula: $\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

21. Identify the equation of the line through $(1,2)$ and $(3,6)$ in slope-intercept form.

    $y=2x$. Slope is $\frac{6-2}{3-1}=2$; passes through origin.

22. Identify the $x$-intercept of $y=2x-8$.

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

23. What is the slope formula for the line through $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run gives the rate of change between two points.

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

    $y-y_1=m(x-x_1)$. Uses a known point and slope to define the line.

25. State the slope-intercept form of a line and identify which value is the $y$-intercept.

    $y=mx+b$; $y$-intercept is $b$. Standard form where $m$ is slope and line crosses $y$-axis at $(0,b)$.

26. What is the $x$-intercept of a graph in terms of the equation $y=f(x)$?

    A solution to $f(x)=0$. The $x$-intercept occurs where the graph crosses the $x$-axis ($y=0$).

27. What is the midpoint formula for endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

    $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the $x$-coordinates and $y$-coordinates separately.

28. What is the axis of symmetry for $y=a(x-h)^2+k$?

    $x=h$. Vertical line through vertex; parabola symmetric about it.

29. Identify the midpoint of the segment with endpoints $(0,8)$ and $(6,2)$.

    $(3,5)$. Midpoint: $\left(\frac{0+6}{2}, \frac{8+2}{2}\right) = (3,5)$.

30. What is the slope of a line perpendicular to a line with slope $m$ (assume $m\ne 0$)?

    $-\frac{1}{m}$. Perpendicular slopes multiply to $-1$.

31. What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$?

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses Pythagorean theorem on the differences in coordinates.

32. Convert $2x-3y=12$ to slope-intercept form $y=mx+b$.

    $y=\frac{2}{3}x-4$. Isolate $y$: $-3y=-2x+12$, then divide by $-3$.

33. What is the relationship between slopes of perpendicular nonvertical lines?

    $m_1m_2=-1$. Their slopes multiply to negative one.

34. What is the distance between $(0,0)$ and $(3,4)$?

    $5$. Distance formula gives $\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$.

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

    $x=4$. Both points have same $x$-coordinate, creating a vertical line.

36. Identify the equation of the line with slope $-2$ passing through $(1,4)$.

    $y=-2x+6$. Using point-slope form: $y - 4 = -2(x - 1)$ simplifies to this.

37. What is the slope of a line perpendicular to a line with slope $m$ (with $m\neq 0$)?

    $-\frac{1}{m}$. Perpendicular slopes multiply to $-1$.

38. 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: change in $y$ divided by change in $x$.

39. What is the slope of a vertical line (for example, $x=-2$)?

    Undefined. Vertical lines have no horizontal change, making slope undefined.

40. Identify the $x$-intercept of the line $2x+3y=12$.

    $(6,0)$. Set $y=0$: $2x+0=12$, so $x=6$.

41. Identify the $y$-intercept of the line $3x+2y=12$.

    $(0,6)$. Set $x=0$ and solve: $2y=12$, so $y=6$.

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

    $y=3x-2$. Substitute slope and point $(0,-2)$ into $y=mx+b$.

43. What is the midpoint formula for the segment joining $(x_1,y_1)$ and $(x_2,y_2)$?

    $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the $x$-coordinates and $y$-coordinates separately.

44. What is the slope-intercept form of a line with slope $m$ and $y$-intercept $b$?

    $y=mx+b$. $m$ is slope, $b$ is where line crosses $y$-axis.

45. What are the coordinates of the origin on a coordinate plane?

    $(0,0)$. The origin is where both axes intersect at zero.

46. Identify the $y$-intercept of the line $2x+3y=12$.

    $(0,4)$. Set $x=0$: $0+3y=12$, so $y=4$.

47. Find the slope of the line through $(2,1)$ and $(6,9)$.

    $2$. $m=\frac{9-1}{6-2}=\frac{8}{4}=2$.

48. What is the slope of a line perpendicular to $y=4x-5$?

    $m=-\frac{1}{4}$. Perpendicular slopes multiply to $-1$: $4 \times (-\frac{1}{4})=-1$.

49. Identify the $x$-intercept of the line $4x-5y=20$.

    $(5,0)$. Set y = 0: $4x = 20$, so $x = 5$.

50. What are the coordinates of the origin on the coordinate plane?

    $(0,0)$. The origin is where the x-axis and y-axis intersect.

51. Identify the distance between $(0,0)$ and $(3,4)$.

    $5$. $\sqrt{(3-0)^2+(4-0)^2}=\sqrt{9+16}=\sqrt{25}=5$.

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

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses Pythagorean theorem on the differences in coordinates.

53. What is the slope relationship of perpendicular nonvertical lines?

    $m_1m_2=-1$. Product of perpendicular slopes always equals negative one.

54. What is the equation of a horizontal line passing through $y=b$?

    $y=b$. Horizontal lines have constant $y$-value for all $x$.

55. What is the midpoint formula for points $(x_1,y_1)$ and $(x_2,y_2)$?

    $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the x-coordinates and y-coordinates.

56. What is the $y$-intercept of a line written in slope-intercept form $y=mx+b$?

    $(0,b)$. The y-intercept occurs where the line crosses the y-axis (when $x=0$).

57. Identify the slope of the line $3x+2y=8$.

    $m=-\frac{3}{2}$. Rewrite as $y=-\frac{3}{2}x+4$ to identify slope.

58. What is the $y$-intercept of a line written as $y=mx+b$?

    $(0,b)$. The y-intercept occurs when x = 0, leaving y = b.

59. What is the slope of a line perpendicular to a line with slope $\frac{3}{4}$?

    $-\frac{4}{3}$. Perpendicular slopes multiply to $-1$: $\frac{3}{4}\cdot(-\frac{4}{3})=-1$.

60. What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$?

    $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Apply the Pythagorean theorem to find straight-line distance.

61. What is the $x$-intercept of the line $2x+3y=6$?

    $(3,0)$. Set $y=0$ and solve: $2x+3(0)=6$, so $x=3$.

62. What is the slope of a horizontal line given by $y=c$?

    $0$. Horizontal lines have no vertical change, so slope is zero.

63. What is the slope of a horizontal line (in simplest form)?

    $0$. No vertical change means rise = 0, so slope = 0.

64. What is the slope of a line perpendicular to a line with slope $\frac{2}{3}$?

    $-\frac{3}{2}$. Take negative reciprocal: $-\frac{1}{\frac{2}{3}}=-\frac{3}{2}$.

65. Identify the vertex of $y=(x-4)^2-9$.

    $(4,-9)$. In vertex form, $(h,k)$ gives the vertex directly.

66. What is the $x$-coordinate of the vertex of $y=2x^2-8x+1$?

    $2$. Use $x=-\frac{b}{2a}$ with $a=2$, $b=-8$: $x=\frac{8}{4}=2$.

67. What is the slope of a horizontal line of the form $y=c$?

    $0$. Horizontal lines have no vertical change, so slope is zero.

68. What is the equation of the line with slope $3$ that passes through $(0,-2)$?

    $y=3x-2$. Point $(0,-2)$ gives $y$-intercept $b=-2$ in $y=mx+b$.

69. A line has slope $\frac{2}{3}$. What is the slope of a perpendicular line?

    $-\frac{3}{2}$. Negative reciprocal: $-1÷\frac{2}{3}=-\frac{3}{2}$.

70. What is the slope of a vertical line written as $x=c$?

    Undefined. Vertical lines have no run, making slope undefined.

71. What is the average rate of change of $f$ from $x=a$ to $x=b$?

    $\frac{f(b)-f(a)}{b-a}$. Slope formula applied to function values at two points.

72. What is the slope-intercept form of a line?

    $y=mx+b$. $m$ is slope and $b$ is the $y$-intercept.

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

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.

74. Identify the slope of a line perpendicular to a line with slope $\frac{2}{3}$.

    $-\frac{3}{2}$. Perpendicular slopes multiply to $-1$: $\frac{2}{3} \cdot (-\frac{3}{2})=-1$.

75. What is the slope of the line $4x-2y=10$?

    $2$. Rewrite as $y=2x-5$, so slope is the coefficient of $x$.

76. Identify the $y$-intercept of the line $4x-5y=10$ as an ordered pair.

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

77. What is the $x$-intercept of the line $y=mx+b$ (as a formula, assuming $m\ne 0$)?

    $x=-\frac{b}{m}$. Set $y=0$ and solve for $x$ to find where line crosses $x$-axis.

78. What is the slope of the line passing through $(2,3)$ and $(6,11)$?

    $2$. Using slope formula: $m=\frac{11-3}{6-2}=\frac{8}{4}=2$.

79. Identify the vertex of the parabola $y=(x-2)^2+5$.

    $(2,5)$. Vertex form $(x-h)^2+k$ has vertex at $(h,k)$.

80. What is the slope of a vertical line given by $x=-3$?

    Undefined. Vertical lines have infinite steepness.

81. What is the $x$-intercept of the line $y=-2x+6$?

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

82. Find the $y$-intercept $b$ of the line $y=3x-7$.

    $-7$. In $y=mx+b$, the constant term $b$ is the y-intercept.

83. Identify the equation of the line through $(1,-2)$ with slope $5$.

    $y=5x-7$. Use point-slope: $y-(-2)=5(x-1)$, then simplify.

84. Identify the equation of the line through $(4,-1)$ with slope $2$.

    $y=2x-9$. Using point-slope form: $y-(-1)=2(x-4)$ simplifies to $y=2x-9$.

85. What is the standard form of a line (with integer coefficients)?

    $Ax+By=C$. $A$, $B$, and $C$ are integers with no common factors.

86. Identify the slope and $y$-intercept of $y=-2x+5$.

    $m=-2,\ b=5$. Read directly from slope-intercept form.

87. What is the slope of a vertical line given by $x=c$?

    Undefined. Vertical lines have zero run, making slope undefined.

88. Identify the $x$-intercept of $y=mx+b$ in terms of $m$ and $b$ (assume $m\ne 0$).

    $\left(-\frac{b}{m},0\right)$. Set $y=0$ and solve for $x$ to find where line crosses $x$-axis.

89. Identify the transformation of $y=f(-x)$ relative to $y=f(x)$.

    Reflect across the $y$-axis. Negative input flips graph across vertical axis.

90. What is the slope of the line passing through $(2,5)$ and $(6,1)$?

    $-1$. Using slope formula: $\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

91. What is the $y$-intercept of a line written as $ax+by=c$ (assume $b\neq 0$)?

    $\left(0,\frac{c}{b}\right)$. Set x = 0 and solve for y to find where line crosses y-axis.

92. Identify the vertex of the parabola $y=(x-3)^2+4$.

    $(3,4)$. Vertex form $y=(x-h)^2+k$ has vertex at $(h,k)$.

93. Identify the slope of the line passing through $(2,5)$ and $(6,1)$.

    $m=-1$. Using slope formula: $\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

94. What are the coordinates of the origin on a coordinate plane?

    $(0,0)$. The origin is where the x-axis and y-axis intersect.

95. Identify the $y$-intercept of $y=-\frac{1}{2}x+6$.

    $(0,6)$. When $x=0$, $y=-\frac{1}{2}(0)+6=6$.

96. What is the midpoint of the segment with endpoints $(-2,5)$ and $(6,1)$?

    $(2,3)$. Apply midpoint formula: $\left(\frac{-2+6}{2},\frac{5+1}{2}\right)=(2,3)$.

97. Identify the transformation of $y=f(x-h)$ relative to $y=f(x)$.

    Shift right $h$ units. Subtracting $h$ from input shifts graph horizontally right.

98. State the formula for the midpoint of a segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$.

    $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the x-coordinates and y-coordinates separately.

99. What is the slope of any vertical line (for example, $x=-3$)?

    Slope is undefined. Vertical lines have no run, making division impossible.

100. What is the slope of a horizontal line given by $y=7$?

     $0$. Horizontal lines have no vertical change.

101. What is the midpoint of the segment with endpoints $(1,-3)$ and $(5,7)$?

     $(3,2)$. Midpoint: $\left(\frac{1+5}{2},\frac{-3+7}{2}\right)=(3,2)$.

102. What is the $y$-intercept of the line $2x+3y=6$?

     $(0,2)$. Set $x=0$ and solve: $2(0)+3y=6$, so $y=2$.

103. What is the $y$-intercept of a line written in slope-intercept form $y=mx+b$?

     $(0,b)$. The $y$-intercept occurs when $x=0$, giving $y=b$.

104. What is the equation of the line with slope $-3$ and $y$-intercept $4$?

     $y=-3x+4$. Substitute $m=-3$ and $b=4$ into slope-intercept form.

105. Find the slope of the line through $(2,5)$ and $(6,1)$.

     $m=-1$. Using slope formula: $\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

106. What is the vertex of $y=(x-4)^2-9$?

     $(4,-9)$. Read vertex directly from form: $(h,k)=(4,-9)$.

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

     $y-y_1=m(x-x_1)$. Shows the line's slope from a specific point.

108. What is the slope of any horizontal line (for example, $y=5$)?

     $m=0$. Horizontal lines have no rise, so slope equals zero.

109. What is the equation of the horizontal line passing through $(0,-3)$?

     $y=-3$. All horizontal lines have form $y=k$ for constant $k$.

110. What is the relationship between slopes of parallel nonvertical lines?

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

111. What is the equation of a vertical line that passes through $x=a$?

     $x=a$. Vertical lines have constant $x$-value for all points.

112. What is the equation of a horizontal line that passes through $y=b$?

     $y=b$. Horizontal lines have constant $y$-value for all points.

113. What does the slope $m$ represent in the line equation $y=mx+b$?

     Rate of change: $m=\frac{\Delta y}{\Delta x}$. Slope measures how much y changes per unit change in x.

114. What is the slope of any vertical line?

     Undefined. Division by zero (no horizontal change) is undefined.

115. What is the distance between the points $(0,0)$ and $(3,4)$?

     $5$. Distance: $\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$.

116. Identify the $y$-intercept of the line $3x+2y=8$.

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

117. What is the equation of the line through $(-2,7)$ and $(3,7)$?

     $y=7$. Both points have same $y$-coordinate, creating a horizontal line.

118. What is the relationship between slopes of parallel nonvertical lines?

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

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

     $4$. Set $x=0$: $3y=12$, so $y=4$ is the $y$-intercept.

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

     $(4,0)$. Set $y=0$ and solve: $3x=12$, so $x=4$.

121. What is the transformation from $y=x^2$ to $y=(x+3)^2-5$?

     Left $3$, down $5$. $(x+3)$ shifts left $3$; $-5$ shifts down $5$.

122. Identify the slope of a line perpendicular to a line with slope $-\frac{2}{5}$.

     $m=\frac{5}{2}$. Negative reciprocal: $-\frac{1}{-\frac{2}{5}}=\frac{5}{2}$.

123. What is the midpoint formula for endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

     $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the $x$-coordinates and $y$-coordinates separately.

124. What are the coordinates of the origin on the coordinate plane?

     $(0,0)$. The origin is where both axes intersect.

125. What is the equation of a line in slope-intercept form?

     $y=mx+b$. Where $m$ is slope and $b$ is the y-intercept.

126. What is the slope of the line $2x-5y=10$?

     $\frac{2}{5}$. Rewrite as $y=\frac{2}{5}x-2$ to find slope.

127. What is the slope of a line parallel to a line with slope $m$?

     $m$. Parallel lines have identical slopes.

128. Identify the distance between $(1,2)$ and $(4,6)$.

     $5$. Distance: $\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9+16} = 5$.

129. What is the relationship between slopes of parallel nonvertical lines?

     Equal slopes: $m_1=m_2$. Parallel lines never intersect.

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

     $y-y_1=m(x-x_1)$. Expresses line using a known point and the slope.

131. What is the slope relationship of parallel nonvertical lines?

     Parallel lines have equal slopes. Same steepness means same slope value.

132. Find the slope of a line perpendicular to $y=\frac{1}{4}x+2$.

     $-4$. Perpendicular to slope $\frac{1}{4}$ is $-\frac{1}{\frac{1}{4}}=-4$.

133. What is the $x$-intercept of a graph in terms of coordinates?

     A point where $y=0$. Graph crosses $x$-axis when height is zero.

134. Identify the equation of the line with slope $2$ passing through $(0,-3)$.

     $y=2x-3$. Use $y=mx+b$ with $m=2$ and $b=-3$.

135. What is the slope of a vertical line of the form $x=c$?

     Undefined. Vertical lines have no horizontal change, so division by zero is undefined.

136. Identify the midpoint of the segment with endpoints $(-2,7)$ and $(6,3)$.

     $(2,5)$. $x=\frac{-2+6}{2}=2$, $y=\frac{7+3}{2}=5$.

137. What is the midpoint formula for endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

     $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the $x$-coordinates and $y$-coordinates separately.

138. What is the equation of the line through $(0,-2)$ and $(4,6)$ in slope-intercept form?

     $y=2x-2$. Slope is $\frac{6-(-2)}{4-0}=2$, passes through $(0,-2)$.

139. What is the slope of a line parallel to $y=-\frac{5}{2}x+9$?

     $-\frac{5}{2}$. Parallel lines share the same slope coefficient.

140. Identify the axis of symmetry of $y=(x+4)^2-1$.

     $x=-4$. Axis passes through vertex; here $h=-4$ in $(x-h)^2$.

141. What are the coordinates of the origin on the coordinate plane?

     $(0,0)$. The origin is where the x-axis and y-axis intersect.

142. What is the slope formula for the line through points $(x_1,y_1)$ and $(x_2,y_2)$?

     $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in y divided by change in x.

143. What is the relationship between slopes of perpendicular nonvertical lines?

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

144. Find the equation of the line with slope $-2$ passing through $(3,4)$.

     $y=-2x+10$. Use point-slope form: $y-4=-2(x-3)$, then simplify.

145. Identify the $x$-intercept of $y=mx+b$ in terms of $m$ and $b$ (assume $m\ne 0$).

     $\left(-\frac{b}{m},0\right)$. Set $y=0$ and solve for $x$: $0=mx+b$ gives $x=-\frac{b}{m}$.

146. What is the midpoint of the segment with endpoints $(-6,4)$ and $(2,10)$?

     $(-2,7)$. Apply midpoint formula: $\left(\frac{-6+2}{2},\frac{4+10}{2}\right)$.

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

     $(0,6)$. Set $x=0$ and solve: $2y=12$, so $y=6$.

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

     $y=mx+b$. $m$ is slope and $b$ is the $y$-intercept.

149. What is the slope of a horizontal line (for example, $y=5$)?

     $0$. Horizontal lines have no vertical change, so slope is zero.

150. What is the axis of symmetry of $y=(x+2)^2-5$?

     $x=-2$. Axis passes through vertex; here $h=-2$.

151. What is the midpoint of the segment with endpoints $(-2,5)$ and $(6,1)$?

     $(2,3)$. Apply midpoint formula: $\left(\frac{-2+6}{2},\frac{5+1}{2}\right)$.

152. What is the relationship between slopes of perpendicular nonvertical lines?

     $m_1m_2=-1$. Their slopes multiply to negative one.

153. Identify the transformation of $y=f(x)+k$ relative to $y=f(x)$.

     Shift up $k$ units. Adding $k$ to output shifts graph vertically upward.

154. What is the $y$-intercept of a line written as $y=mx+b$?

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

155. Identify the equation of the line with slope $-3$ and $y$-intercept $4$.

     $y=-3x+4$. Direct substitution into slope-intercept form.

156. What is the vertex of the parabola $y=(x-3)^2-5$?

     $(3,-5)$. Vertex form $y = (x-h)^2 + k$ has vertex at $(h,k)$.

157. What is the $y$-intercept of the line $3x+2y=8$?

     $(0,4)$. Set $x=0$: $3(0)+2y=8$, so $y=4$.

158. What is the $x$-intercept of a graph in terms of the equation value of $y$?

     The point where $y=0$. Where the graph crosses the x-axis.

159. What is the vertex form of a quadratic function with vertex $(h,k)$?

     $y=a(x-h)^2+k$. Shows parabola with vertex at $(h,k)$ and vertical stretch $a$.

160. Identify whether the relation $x^2+y^2=9$ is a function of $x$ using the vertical line test.

     Not a function. Circle equation fails vertical line test; some $x$ values have two $y$ values.

161. What is the slope of any horizontal line?

     $0$. Horizontal lines have no vertical change, so slope is zero.

162. What is the vertex form of a quadratic function with vertex $(h,k)$?

     $y=a(x-h)^2+k$. Shows parabola with vertex at $(h,k)$ and width factor $a$.

163. What is the slope of a line perpendicular to a line with slope $\frac{2}{3}$?

     $-\frac{3}{2}$. Perpendicular slopes multiply to $-1$: $\frac{2}{3}\times(-\frac{3}{2})=-1$.

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

     $y=k$. Horizontal lines have constant y-values.

165. What is the slope of the line $2x+3y=12$?

     $-\frac{2}{3}$. Rewrite as $y=-\frac{2}{3}x+4$ to find slope.

166. What is the y-intercept of a graph defined as?

     The point where $x=0$. Where the graph crosses the y-axis.

167. What is the vertex of the parabola $y=(x-2)^2-9$?

     $(2,-9)$. Vertex form $y=a(x-h)^2+k$ has vertex at $(h,k)$.

168. What is the slope of the line through $(2,5)$ and $(6,1)$?

     $-1$. Using slope formula: $\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

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

     $x=a$. Vertical lines have constant $x$-value for all $y$.

170. Find the equation of the line with slope $-3$ passing through $(1,5)$ in slope-intercept form.

     $y=-3x+8$. Use $y-5=-3(x-1)$, then simplify to slope-intercept form.

171. What is the equation of a vertical line passing through $x=k$?

     $x=k$. Vertical lines have constant x-values.

172. Identify the $x$-intercept of the line $3x+2y=12$.

     $(4,0)$. Set $y=0$ and solve: $3x=12$, so $x=4$.

173. What is the $y$-intercept of a graph in terms of coordinates?

     A point where $x=0$. Graph crosses $y$-axis when horizontal position is zero.

174. What is the slope of a vertical line?

     Undefined. Division by zero (no horizontal change) is undefined.

175. What are the coordinates of the origin on the coordinate plane?

     $(0,0)$. The origin is where both axes intersect at zero.

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

     $(0,b)$. The point where the line crosses the $y$-axis when $x=0$.

177. Identify the slope of the line $4x-2y=10$.

     $2$. Rewrite as $y=2x-5$; slope is coefficient of $x$.

178. What is the equation of the line with slope $-3$ and $y$-intercept $5$?

     $y=-3x+5$. Substitute slope and intercept directly into slope-intercept form.

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

     $y=3x-2$. Use $y=mx+b$ with $m=3$ and $b=-2$.

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

     $-\frac{5}{2}$. Perpendicular slopes multiply to -1: $\frac{2}{5} \times (-\frac{5}{2}) = -1$.

181. What is the slope of a line parallel to a line with slope $-5$?

     $-5$. Parallel lines have identical slopes.

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

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

183. What is the equation of the line through $(4,-1)$ with slope $-2$ in slope-intercept form?

     $y=-2x+7$. Use point-slope form then convert: $y-(-1)=-2(x-4)$ gives $y=-2x+7$.

184. What is the slope of the line passing through $(2,3)$ and $(6,11)$?

     $2$. Use slope formula: $\frac{11-3}{6-2}=\frac{8}{4}=2$.

185. What is the axis of symmetry of the parabola $y=(x+2)^2-7$?

     $x=-2$. Axis of symmetry is $x=h$ for parabola $y=(x-h)^2+k$.

186. For $y=a(x-h)^2+k$, what is the axis of symmetry?

     $x=h$. The parabola is symmetric about the vertical line through the vertex.

187. Identify the midpoint of the segment with endpoints $( -2,4)$ and $(6,0)$.

     $(2,2)$. $\left(\frac{-2+6}{2},\frac{4+0}{2}\right)=(2,2)$

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

     $y=3x-2$. Substitute $m=3$ and $b=-2$ into slope-intercept form.

189. What is the slope of a line parallel to a line with slope $-\frac{5}{2}$?

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

190. What is the slope of a horizontal line?

     $0$. No vertical change means zero rise over run.

191. Identify the equation of the line with slope $3$ and y-intercept $-2$.

     $y=3x-2$. Substitute slope and y-intercept into $y=mx+b$.

192. What is the slope of a horizontal line written as $y=c$?

     $0$. Horizontal lines have no rise, so slope equals zero.

193. Identify the slope of the line $4x-2y=10$.

     $2$. Rewrite as $y=2x-5$; coefficient of $x$ is slope = 2.

194. Identify the $x$-intercepts of $y=(x-2)(x+3)$.

     $(-3,0)$ and $(2,0)$. Set $y=0$: factors give $x=2$ and $x=-3$.

195. State the slope formula for 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 two points gives the slope.

196. State the point-slope form of a line with slope $m$ through $(x_1,y_1)$.

     $y-y_1=m(x-x_1)$. Rearranges slope formula to isolate y-difference.

197. What is the x-intercept of a graph defined as?

     The point where $y=0$. Where the graph crosses the x-axis.

198. What is the equation of the line through $(1,-2)$ with slope $5$ in slope-intercept form?

     $y=5x-7$. Use point-slope: $y-(-2)=5(x-1)$, then simplify.

199. What are the $x$- and $y$-intercepts of $2x+3y=6$?

     $x$-int $(3,0)$; $y$-int $(0,2)$. Set $y=0$ for $x$-int; set $x=0$ for $y$-int.

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