Table & Graph Interpretation - ACT Math

1. What does a negative slope indicate about the relationship between $x$ and $y$?

   As $x$ increases, $y$ decreases. Negative slope means the line falls from left to right.

2. What is the median of the dataset: 3, 7, 9, 15, 18?

   9. The middle value when data is ordered from least to greatest.

3. Find the mode in the dataset: 2, 4, 4, 6, 8.

   The mode is 4. Mode is the value that appears most frequently.

4. What feature of a line graph shows a decrease in the dependent variable?

   Negative slope. Negative slope indicates the line falls from left to right.

5. What does a positive slope indicate about the relationship between $x$ and $y$?

   As $x$ increases, $y$ increases. Positive slope means the line rises from left to right.

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

   Undefined slope. Same $x$-values mean vertical line with undefined slope.

7. If $f(x) = 2x - 1$, find $f^{-1}(3)$.

   2. Find the input value that gives output 3: $2x - 1 = 3$, so $x = 2$.

8. Calculate the midpoint between points (2, 3) and (4, 7).

   (3, 5). Midpoint formula: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) = (3, 5)$.

9. Identify $f(3)$ if a table lists $(3, -2)$ as an ordered pair for $(x,f(x))$.

   $-2$. The table directly gives the function value.

10. Determine the range of the dataset: 2, 4, 6, 8, 10.

    8. Range equals the maximum value minus the minimum value: $10 - 2 = 8$.

11. Which graph type is best for showing proportions in different categories?

    A pie chart. Pie charts show how parts make up a whole.

12. Identify the $x$-intercept of the line $y=4x+8$.

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

13. Which graph type is best for comparing multiple datasets?

    A grouped bar chart. Shows multiple categories side by side for comparison.

14. Identify the y-intercept in the equation $y = 3x + 7$.

    The y-intercept is 7. The constant term in slope-intercept form $y = mx + b$.

15. What does an undefined slope indicate about a line on a coordinate plane?

    The line is vertical. Infinite slope means no horizontal change occurs.

16. What graph is most appropriate for showing trends over time?

    Line graph. Line graphs connect data points to show changes over time periods.

17. Calculate the sum of the y-coordinates for points (3, 5) and (4, 9).

    14. Add the y-coordinates: $5 + 9 = 14$.

18. What does a line of best fit indicate in a scatter plot?

    It indicates the general direction or trend of the data. Best-fitting line shows overall pattern in data.

19. Determine the y-intercept of the quadratic $y = x^2 - 4x + 4$.

    4. Y-intercept occurs when $x = 0$: $y = 0^2 - 4(0) + 4 = 4$.

20. What is the correlation coefficient range?

    -1 to 1. Correlation coefficients measure linear association strength from -1 to 1.

21. Which data measure is least affected by extreme values?

    Median. Median is resistant to outliers, unlike mean which is affected by extremes.

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

    $\frac{f(b)-f(a)}{b-a}$. Total change in output divided by change in input.

23. What does slope $m=0$ indicate about a line on a coordinate plane?

    The line is horizontal. Zero slope means no vertical change occurs.

24. Identify the outlier in the dataset: 5, 7, 8, 9, 50.

    The outlier is 50. Value significantly different from other data points.

25. What does the $y$-intercept represent on a graph of $y$ versus $x$?

    The value of $y$ when $x=0$. Where the line crosses the vertical axis.

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

    $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Pythagorean theorem applied to coordinate differences.

27. Identify the $y$-intercept of the line $y=3x-7$.

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

28. Identify the quadrant where point $(-3, 4)$ is located.

    Quadrant II. Quadrant II contains points with negative x and positive y coordinates.

29. What does it mean if a line passes through the origin on a $y$ versus $x$ graph?

    The $y$-intercept is $0$. The line passes through point $(0,0)$.

30. What is the meaning of a data point $(x,y)$ on a graph of $y$ versus $x$?

    When $x$ equals the first value, $y$ equals the second. The input value determines the output value.

31. What is the dependent variable on a typical graph of $y$ versus $x$?

    $y$ (vertical axis). The output variable that responds to input changes.

32. What is the definition of a histogram?

    A histogram is a bar graph representing the frequency distribution of data. Shows frequency distribution using adjacent bars.

33. A table shows points $(1,4)$ and $(4,10)$. Identify the slope.

    $2$. $m = \frac{10-4}{4-1} = \frac{6}{3} = 2$

34. What is the mode of the data set: 8, 8, 9, 10, 10, 10?

    10. The mode is the value that appears most frequently in the dataset.

35. A line increases $12$ units in $y$ while increasing $3$ units in $x$. Identify the slope.

    $4$. $m = \frac{\Delta y}{\Delta x} = \frac{12}{3} = 4$

36. Identify the x-intercept in the equation $y = -2x + 10$.

    The x-intercept is 5. Where the line crosses the x-axis when $y = 0$.

37. Identify the symmetry of the function $y = x^2$.

    Symmetric about the y-axis. Even functions satisfy $f(-x) = f(x)$, creating y-axis symmetry.

38. Find the slope of the line passing through points (2, 3) and (4, 7).

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

39. What is the interquartile range of the dataset: 1, 3, 5, 7, 9, 11?

    6. IQR is $Q_3 - Q_1 = 9 - 3 = 6$ for this ordered dataset.

40. What is the percent change from $A$ to $B$?

    $\frac{B-A}{A}\times 100\%$. Difference divided by original, times 100.

41. What does a correlation coefficient of 0 indicate?

    No correlation. A correlation of 0 means no linear relationship exists between variables.

42. Determine the y-intercept in the function $f(x) = 4x - 5$.

    The y-intercept is -5. The constant term when $x = 0$.

43. Identify the midpoint of the segment with endpoints $(1,7)$ and $(5,3)$.

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

44. Given points (2, 3) and (2, 7), what is the slope of the line?

    Undefined. Vertical lines have undefined slope due to division by zero.

45. 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.

46. What is the formula for the slope of a line given 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 on the line.

47. Identify the value of $x$ when $y=0$ for the line $y=-2x+6$.

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

48. What is the primary use of a scatter plot?

    To show the relationship between two variables. Points plotted to reveal correlations between variables.

49. Determine the mode in the dataset: 3, 3, 6, 9, 9, 9.

    The mode is 9. Value 9 appears three times, more than others.

50. Identify the distance between $(0,0)$ and $(6,8)$.

    $10$. $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = 10$

51. What is the definition of standard deviation?

    Standard deviation measures the amount of variation or dispersion in a dataset. Quantifies how spread out data values are.

52. What is the percent change from $50$ to $65$?

    $30\%$. $\frac{65-50}{50} \times 100\% = 30\%$

53. Identify the range in the dataset: 4, 7, 9, 10, 15.

    The range is 11. Range equals maximum minus minimum: $15 - 4 = 11$.

54. What does it mean if a graph is decreasing on an interval?

    As $x$ increases, $y$ decreases. The function has negative slope on that interval.

55. Identify the median in the dataset: 3, 5, 7, 9, 11.

    The median is 7. Middle value when data is ordered from least to greatest.

56. For $y = mx + b$, what does $b$ represent?

    Y-intercept. In slope-intercept form, $b$ is the point where the line crosses the y-axis.

57. In a linear relationship, what does the y-intercept represent?

    Initial value. The y-intercept represents the starting value when $x = 0$.

58. Identify the slope between $(2,5)$ and $(6,13)$.

    $m=2$. $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$

59. What does a box plot visually represent?

    A box plot represents the data's minimum, first quartile, median, third quartile, and maximum. Shows five-number summary of data distribution.

60. What is the standard deviation a measure of in a data set?

    Data dispersion. Standard deviation quantifies how spread out data points are from the mean.

61. 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 separately.

62. Identify the line equation form for a vertical line through $(4, 3)$.

    $x = 4$. Vertical lines have constant x-coordinates and undefined slope.

63. What is a bar graph used for?

    To compare quantities across different categories. Bars represent discrete categories for comparison.

64. Identify the type of variation: $y = kx$.

    Direct variation. Direct variation means $y$ is proportional to $x$ with constant $k$.

65. What is the mean of the data set: 5, 10, 15, 20, 25?

    15. Sum all values and divide by the count: $(5+10+15+20+25)/5 = 15$.

66. Find the mean of the dataset: 5, 15, 25, 35.

    The mean is 20. Sum divided by count: $(5+15+25+35)/4 = 20$.

67. Identify the median in the dataset: 1, 3, 3, 6, 7, 8, 9.

    The median is 6. Middle value in ordered list of 7 values.

68. What is the independent variable on a typical graph of $y$ versus $x$?

    $x$ (horizontal axis). The input variable that can be controlled.

69. If a line passes through points (1, 2) and (3, 6), what is the slope of the line?

    2. Slope formula: $m = \frac{y_2-y_1}{x_2-x_1} = \frac{6-2}{3-1} = 2$.

70. Find the mean of the dataset: 10, 20, 30, 40.

    The mean is 25. Sum of values divided by count: $(10+20+30+40)/4 = 25$.

71. Find the slope of the line passing through points $(2, 3)$ and $(4, 7)$.

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

72. Identify the slope of the line represented by the equation $4x - 2y = 8$.

    2. Rearrange to $y = 2x - 4$; the slope is the coefficient of $x$.

73. Which graph feature indicates a direct correlation between two variables?

    Positive slope. Positive slope means as one variable increases, the other increases too.

74. Determine the mode in the dataset: 6, 6, 7, 8, 9.

    The mode is 6. The most frequently occurring value.

75. Find the range in the dataset: 2, 8, 10, 15, 24.

    The range is 22. Maximum minus minimum: $24 - 2 = 22$.

76. Which type of graph is best used to display categorical data?

    Bar graph. Bar graphs effectively compare distinct categories side by side.

77. What does it mean if a graph is increasing on an interval?

    As $x$ increases, $y$ increases. The function has positive slope on that interval.

78. What does it mean if a graph is constant on an interval?

    $y$ stays the same as $x$ changes. The function has zero slope on that interval.

79. What is the average rate of change of $f$ from $x=1$ to $x=5$ if $f(1)=2$ and $f(5)=14$?

    $3$. $\frac{14-2}{5-1} = \frac{12}{4} = 3$

80. Identify the y-intercept of the line $y = -5x + 4$.

    4. In slope-intercept form $y = mx + b$, the constant term is the y-intercept.

81. A table shows $x:0,1,2$ and $y:5,8,11$. What is the $y$-intercept?

    $5$. The $y$-intercept occurs when $x = 0$.

82. Which graph best shows a frequency distribution of a data set?

    Histogram. Histograms use bars to display the frequency of data intervals.

83. Determine the median of the dataset: 12, 15, 18, 22, 30.

    The median is 18. Middle value in the ordered dataset.

84. Find the slope of a line perpendicular to $y = \frac{1}{3}x + 1$.

    -3. Perpendicular lines have slopes that are negative reciprocals.

85. Calculate the average rate of change of $y = x^2$ from $x=1$ to $x=3$.

    4. Average rate of change: $\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4$.

86. What is the primary feature of a pie chart?

    A pie chart shows parts of a whole as slices of a circle. Circular representation of proportional data.

87. What does the correlation coefficient indicate?

    It indicates the strength and direction of a linear relationship between variables. Ranges from -1 to 1, measuring linear association.

88. What type of chart is useful for showing parts of a whole?

    Pie chart. Pie charts visually represent proportions of a total as sectors.

89. A table shows $x:2,3,4$ and $y:9,12,15$. Identify the rule in slope-intercept form.

    $y=3x+3$. Slope is $3$, $y$-intercept is $3$: $y = 3x + 3$.

90. Which plot displays the five-number summary of a data set?

    Box plot. Box plots show minimum, Q1, median, Q3, and maximum values.

91. A graph shows a line with slope $-3$ passing through $(0,2)$. What is its equation?

    $y=-3x+2$. Use slope-intercept form: $y = mx + b$.

92. What is the y-coordinate of the point where the line $y = 3x + 2$ intersects the y-axis?

    2. The y-intercept occurs when $x = 0$, so $y = 3(0) + 2 = 2$.

93. Identify the missing $y$ value for a linear table: $(1,2)$, $(3,6)$, $(5,\, ?)$.

    $10$. Linear pattern: slope is $2$, so $y = 2 + 2(4) = 10$.

94. What does the slope represent in a linear graph?

    The slope represents the rate of change between variables. Slope shows how $y$ changes as $x$ increases by one unit.

95. What is the equation of a horizontal line passing through $(7, -2)$?

    $y = -2$. Horizontal lines have constant y-coordinates and zero slope.

96. What is the definition of the mean in a dataset?

    The mean is the sum of all data points divided by the number of points. This is the standard statistical definition.

97. What does the $x$-intercept represent on a graph of $y$ versus $x$?

    The value of $x$ when $y=0$. Where the line crosses the horizontal axis.

98. Which graph type is best for showing trends over time?

    A line graph. Line graphs effectively display data changes over time.

99. What does the slope of a line represent in a distance-time graph?

    Speed. Slope represents the rate of change, which is speed in distance-time graphs.

100. In a scatter plot, what does a cluster of points indicate?

     Possible correlation. Clustered points suggest a relationship between the two variables.

101. Identify the slope of a line through $(0,4)$ and $(3,4)$.

     $0$. Same $y$-values mean horizontal line with zero slope.

102. Identify the average rate of change from $(2,9)$ to $(8,-3)$.

     $-2$. $\frac{-3-9}{8-2} = \frac{-12}{6} = -2$

103. Given the function $f(x) = 3x + 7$, what is $f(2)$?

     13. Substitute $x = 2$: $f(2) = 3(2) + 7 = 13$.

104. What is the vertex of the parabola $y = -x^2 + 4x - 3$?

     (2, 1). For $y = ax^2 + bx + c$, vertex is at $x = -\frac{b}{2a} = 2$.

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