Table and Graph Interpretation - ISEE Middle Level: Mathematics Achievement
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A table shows $(x,y)$ pairs: $(0,7)$, $(1,5)$, $(2,3)$. What is the change in $y$ per $+1$ in $x$?
A table shows $(x,y)$ pairs: $(0,7)$, $(1,5)$, $(2,3)$. What is the change in $y$ per $+1$ in $x$?
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$-2$ per $1$ unit of $x$. Each unit increase in x results in a consistent decrease of 2 in y, indicating a constant rate.
$-2$ per $1$ unit of $x$. Each unit increase in x results in a consistent decrease of 2 in y, indicating a constant rate.
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A line graph passes through $(2,8)$ and $(6,20)$. What is the average rate of change?
A line graph passes through $(2,8)$ and $(6,20)$. What is the average rate of change?
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$3$. The average rate of change is the slope between the points, calculated as rise over run.
$3$. The average rate of change is the slope between the points, calculated as rise over run.
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A bar chart has counts $A=12$, $B=9$, $C=15$. Which category has the greatest value?
A bar chart has counts $A=12$, $B=9$, $C=15$. Which category has the greatest value?
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$C$. Comparing the bar heights shows C with the tallest bar, indicating the highest count.
$C$. Comparing the bar heights shows C with the tallest bar, indicating the highest count.
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A bar chart has values $A=18$, $B=6$, $C=12$. What is the difference between the greatest and least values?
A bar chart has values $A=18$, $B=6$, $C=12$. What is the difference between the greatest and least values?
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$12$. Subtracting the smallest value (6) from the largest (18) gives the range of the data.
$12$. Subtracting the smallest value (6) from the largest (18) gives the range of the data.
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A pie chart shows $\frac{1}{4}$ of students walk to school. What percent is this?
A pie chart shows $\frac{1}{4}$ of students walk to school. What percent is this?
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$25%$. Converting the fraction 1/4 to a percentage by multiplying by 100 yields the result.
$25%$. Converting the fraction 1/4 to a percentage by multiplying by 100 yields the result.
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A pie chart shows $40%$ of a budget is rent. What fraction of the budget is rent in simplest form?
A pie chart shows $40%$ of a budget is rent. What fraction of the budget is rent in simplest form?
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$\frac{2}{5}$. Simplifying 40/100 by dividing numerator and denominator by 20 produces the fraction in lowest terms.
$\frac{2}{5}$. Simplifying 40/100 by dividing numerator and denominator by 20 produces the fraction in lowest terms.
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Identify the independent variable on a standard graph where $x$ is time and $y$ is distance.
Identify the independent variable on a standard graph where $x$ is time and $y$ is distance.
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Time (the $x$-variable). The independent variable, typically time on the x-axis, does not depend on other factors in the relationship.
Time (the $x$-variable). The independent variable, typically time on the x-axis, does not depend on other factors in the relationship.
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Identify the dependent variable on a standard graph where $x$ is time and $y$ is distance.
Identify the dependent variable on a standard graph where $x$ is time and $y$ is distance.
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Distance (the $y$-variable). The dependent variable, often distance on the y-axis, varies based on changes in the independent variable.
Distance (the $y$-variable). The dependent variable, often distance on the y-axis, varies based on changes in the independent variable.
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Find the slope between $(2,5)$ and $(6,13)$.
Find the slope between $(2,5)$ and $(6,13)$.
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$m=2$. Applying the slope formula gives a rise of 8 over a run of 4, resulting in the calculated value.
$m=2$. Applying the slope formula gives a rise of 8 over a run of 4, resulting in the calculated value.
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Find the slope between $( -1,4)$ and $(3,-8)$.
Find the slope between $( -1,4)$ and $(3,-8)$.
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$m=-3$. The slope formula yields a change in y of -12 divided by a change in x of 4.
$m=-3$. The slope formula yields a change in y of -12 divided by a change in x of 4.
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What is the $y$-intercept of the line through $(0,-3)$ and $(4,5)$?
What is the $y$-intercept of the line through $(0,-3)$ and $(4,5)$?
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$-3$. The point (0, -3) directly provides the y-intercept as it lies on the y-axis.
$-3$. The point (0, -3) directly provides the y-intercept as it lies on the y-axis.
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What is the $x$-intercept of the line $y=2x-10$?
What is the $x$-intercept of the line $y=2x-10$?
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$x=5$. Setting y=0 in the equation and solving for x determines the x-intercept.
$x=5$. Setting y=0 in the equation and solving for x determines the x-intercept.
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What does the maximum value on a graph represent in the context of the data?
What does the maximum value on a graph represent in the context of the data?
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The greatest observed $y$-value (output). The maximum y-value identifies the peak output observed in the dataset represented by the graph.
The greatest observed $y$-value (output). The maximum y-value identifies the peak output observed in the dataset represented by the graph.
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What does the minimum value on a graph represent in the context of the data?
What does the minimum value on a graph represent in the context of the data?
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The smallest observed $y$-value (output). The minimum y-value marks the lowest output point in the data depicted on the graph.
The smallest observed $y$-value (output). The minimum y-value marks the lowest output point in the data depicted on the graph.
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What does the range of a graph’s $y$-values mean for the data shown?
What does the range of a graph’s $y$-values mean for the data shown?
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All output values $y$ that occur on the graph. The range encompasses all possible output values produced by the function over the graphed interval.
All output values $y$ that occur on the graph. The range encompasses all possible output values produced by the function over the graphed interval.
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What does the domain of a graph’s $x$-values mean for the data shown?
What does the domain of a graph’s $x$-values mean for the data shown?
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All input values $x$ for which the graph has points. The domain includes all input values for which the function is defined and produces outputs on the graph.
All input values $x$ for which the graph has points. The domain includes all input values for which the function is defined and produces outputs on the graph.
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What is the average rate of change of $y$ from $x=a$ to $x=b$ on a graph?
What is the average rate of change of $y$ from $x=a$ to $x=b$ on a graph?
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$\frac{y(b)-y(a)}{b-a}$. The average rate of change measures the overall slope of the secant line connecting the points at x=a and x=b.
$\frac{y(b)-y(a)}{b-a}$. The average rate of change measures the overall slope of the secant line connecting the points at x=a and x=b.
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A table shows $(x,y)$ pairs: $(1,3)$, $(2,6)$, $(3,9)$. What is $y$ when $x=4$?
A table shows $(x,y)$ pairs: $(1,3)$, $(2,6)$, $(3,9)$. What is $y$ when $x=4$?
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$12$. The pattern y=3x extends linearly to predict the value at x=4.
$12$. The pattern y=3x extends linearly to predict the value at x=4.
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What does the $x$-intercept represent on a graph of $y$ versus $x$?
What does the $x$-intercept represent on a graph of $y$ versus $x$?
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The value of $x$ when $y=0$. The x-intercept is the point where the graph crosses the x-axis, indicating the input value that yields zero output.
The value of $x$ when $y=0$. The x-intercept is the point where the graph crosses the x-axis, indicating the input value that yields zero output.
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What does the $y$-intercept represent on a graph of $y$ versus $x$?
What does the $y$-intercept represent on a graph of $y$ versus $x$?
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The value of $y$ when $x=0$. The y-intercept is the point where the graph intersects the y-axis, showing the output when the input is zero.
The value of $y$ when $x=0$. The y-intercept is the point where the graph intersects the y-axis, showing the output when the input is zero.
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What does a positive slope on a line graph indicate about $y$ as $x$ increases?
What does a positive slope on a line graph indicate about $y$ as $x$ increases?
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$y$ increases as $x$ increases. A positive slope reflects a direct relationship where the dependent variable rises with increases in the independent variable.
$y$ increases as $x$ increases. A positive slope reflects a direct relationship where the dependent variable rises with increases in the independent variable.
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What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?
What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?
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$m=\frac{y_2-y_1}{x_2-x_1}$. The slope formula computes the rate of change as the difference in y-values divided by the difference in x-values between two points.
$m=\frac{y_2-y_1}{x_2-x_1}$. The slope formula computes the rate of change as the difference in y-values divided by the difference in x-values between two points.
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What do the coordinates $(x,y)$ of a point on a graph represent?
What do the coordinates $(x,y)$ of a point on a graph represent?
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$x$ is the input (horizontal); $y$ is the output (vertical). In coordinate geometry, the x-coordinate denotes the independent variable on the horizontal axis, while y denotes the dependent variable on the vertical axis.
$x$ is the input (horizontal); $y$ is the output (vertical). In coordinate geometry, the x-coordinate denotes the independent variable on the horizontal axis, while y denotes the dependent variable on the vertical axis.
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What does a horizontal line ($m=0$) mean about the relationship between $x$ and $y$?
What does a horizontal line ($m=0$) mean about the relationship between $x$ and $y$?
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$y$ is constant for all $x$ values. A zero slope signifies that the dependent variable remains unchanged regardless of variations in the independent variable.
$y$ is constant for all $x$ values. A zero slope signifies that the dependent variable remains unchanged regardless of variations in the independent variable.
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What does a negative slope on a line graph indicate about $y$ as $x$ increases?
What does a negative slope on a line graph indicate about $y$ as $x$ increases?
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$y$ decreases as $x$ increases. A negative slope indicates an inverse relationship where the dependent variable falls as the independent variable rises.
$y$ decreases as $x$ increases. A negative slope indicates an inverse relationship where the dependent variable falls as the independent variable rises.
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