Function Rules From Data - ISEE Upper Level: Mathematics Achievement
Card 1 of 25
Identify the rule from the table: $(x,y)$ are $(1,4)$, $(2,7)$, $(3,10)$, $(4,13)$. What is $y$?
Identify the rule from the table: $(x,y)$ are $(1,4)$, $(2,7)$, $(3,10)$, $(4,13)$. What is $y$?
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$y=3x+1$. The table exhibits constant first differences of 3 in $y$, confirming a linear rule with slope 3 and fitting the points.
$y=3x+1$. The table exhibits constant first differences of 3 in $y$, confirming a linear rule with slope 3 and fitting the points.
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Identify the rule from the table: $(x,y)$ are $(2,10)$, $(4,20)$, $(6,30)$. What is $y$?
Identify the rule from the table: $(x,y)$ are $(2,10)$, $(4,20)$, $(6,30)$. What is $y$?
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$y=5x$. The points show direct proportionality with a constant ratio of 5, yielding this linear rule through the origin.
$y=5x$. The points show direct proportionality with a constant ratio of 5, yielding this linear rule through the origin.
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Identify the rule from the table: $(x,y)$ are $(-2,4)$, $(-1,1)$, $(0,0)$, $(1,1)$, $(2,4)$. What is $y$?
Identify the rule from the table: $(x,y)$ are $(-2,4)$, $(-1,1)$, $(0,0)$, $(1,1)$, $(2,4)$. What is $y$?
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$y=x^2$. This symmetric set of points around the vertex at (0,0) matches the standard quadratic form.
$y=x^2$. This symmetric set of points around the vertex at (0,0) matches the standard quadratic form.
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Identify the rule from the table: $(x,y)$ are $(1,3)$, $(2,6)$, $(3,12)$, $(4,24)$. What is $y$?
Identify the rule from the table: $(x,y)$ are $(1,3)$, $(2,6)$, $(3,12)$, $(4,24)$. What is $y$?
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$y=3\cdot 2^{x-1}$. The y-values double each time x increases by 1, fitting an exponential model with base 2 adjusted for the initial value.
$y=3\cdot 2^{x-1}$. The y-values double each time x increases by 1, fitting an exponential model with base 2 adjusted for the initial value.
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Which table pattern indicates an inverse variation rule $y=\frac{k}{x}$?
Which table pattern indicates an inverse variation rule $y=\frac{k}{x}$?
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Constant product $xy=k$. Inverse variation produces a constant product of x and y across the table entries.
Constant product $xy=k$. Inverse variation produces a constant product of x and y across the table entries.
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Identify the rule from the table: $(x,y)$ are $(1,8)$, $(2,4)$, $(4,2)$, $(8,1)$. What is $y$?
Identify the rule from the table: $(x,y)$ are $(1,8)$, $(2,4)$, $(4,2)$, $(8,1)$. What is $y$?
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$y=\frac{8}{x}$. The product $xy=8$ is constant, indicating inverse variation with this form fitting all points.
$y=\frac{8}{x}$. The product $xy=8$ is constant, indicating inverse variation with this form fitting all points.
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Identify the rule from the table: $(x,y)$ are $(0,2)$, $(1,6)$, $(2,18)$, $(3,54)$. What is $y$?
Identify the rule from the table: $(x,y)$ are $(0,2)$, $(1,6)$, $(2,18)$, $(3,54)$. What is $y$?
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$y=2\cdot 3^x$. The y-values multiply by 3 each time x increases by 1, matching an exponential with base 3 and initial factor 2.
$y=2\cdot 3^x$. The y-values multiply by 3 each time x increases by 1, matching an exponential with base 3 and initial factor 2.
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Which table pattern indicates an exponential rule when $x$ increases by $1$ each step?
Which table pattern indicates an exponential rule when $x$ increases by $1$ each step?
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Constant ratio $\frac{y_{k+1}}{y_k}$. Exponential functions maintain a constant ratio between consecutive y-values for equal x increments.
Constant ratio $\frac{y_{k+1}}{y_k}$. Exponential functions maintain a constant ratio between consecutive y-values for equal x increments.
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What is the general rule for a linear function written in slope-intercept form?
What is the general rule for a linear function written in slope-intercept form?
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$y=mx+b$. This form expresses a linear function with slope $m$ and $y$-intercept $b$.
$y=mx+b$. This form expresses a linear function with slope $m$ and $y$-intercept $b$.
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What is the slope formula for a line through points $(x_1,y_1)$ and $(x_2,y_2)$?
What is the slope formula for a line through points $(x_1,y_1)$ and $(x_2,y_2)$?
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$m=\frac{y_2-y_1}{x_2-x_1}$. This formula computes the rate of change as the difference in $y$-coordinates divided by the difference in $x$-coordinates between two points on the line.
$m=\frac{y_2-y_1}{x_2-x_1}$. This formula computes the rate of change as the difference in $y$-coordinates divided by the difference in $x$-coordinates between two points on the line.
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Identify the rule: a line has slope $-4$ and $y$-intercept $6$. What is the equation?
Identify the rule: a line has slope $-4$ and $y$-intercept $6$. What is the equation?
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$y=-4x+6$. The slope-intercept form directly incorporates the given slope and $y$-intercept into the linear equation.
$y=-4x+6$. The slope-intercept form directly incorporates the given slope and $y$-intercept into the linear equation.
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Identify the rule: a line passes through $(3,-2)$ and has slope $5$. What is $y$ in terms of $x$?
Identify the rule: a line passes through $(3,-2)$ and has slope $5$. What is $y$ in terms of $x$?
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$y=5x-17$. Using point-slope form with the given point and slope yields this equation after simplification.
$y=5x-17$. Using point-slope form with the given point and slope yields this equation after simplification.
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Identify the rule from the table: $(x,y)$ are $(0,5)$, $(1,2)$, $(2,-1)$, $(3,-4)$. What is $y$?
Identify the rule from the table: $(x,y)$ are $(0,5)$, $(1,2)$, $(2,-1)$, $(3,-4)$. What is $y$?
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$y=-3x+5$. The table shows constant first differences of -3 in $y$, indicating a slope of -3 and $y$-intercept of 5.
$y=-3x+5$. The table shows constant first differences of -3 in $y$, indicating a slope of -3 and $y$-intercept of 5.
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What is slope in a table when $x$ increases by $1$ each row (in terms of $\Delta y$)?
What is slope in a table when $x$ increases by $1$ each row (in terms of $\Delta y$)?
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$m=\Delta y$ per $1$ unit of $x$. When $x$ increases by 1 unit per row, the slope equals the constant change in $y$ for each step.
$m=\Delta y$ per $1$ unit of $x$. When $x$ increases by 1 unit per row, the slope equals the constant change in $y$ for each step.
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Identify the rule: points $(0,3)$ and $(2,7)$ lie on a line. What is $y$ in terms of $x$?
Identify the rule: points $(0,3)$ and $(2,7)$ lie on a line. What is $y$ in terms of $x$?
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$y=2x+3$. The slope calculated from the points is 2, and the $y$-intercept is 3, yielding the linear equation.
$y=2x+3$. The slope calculated from the points is 2, and the $y$-intercept is 3, yielding the linear equation.
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What is the point-slope form of a line with slope $m$ passing through $(x_1,y_1)$?
What is the point-slope form of a line with slope $m$ passing through $(x_1,y_1)$?
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$y-y_1=m(x-x_1)$. This form derives from the slope formula, expressing the line using a known point and slope.
$y-y_1=m(x-x_1)$. This form derives from the slope formula, expressing the line using a known point and slope.
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Which condition in a table confirms a linear rule when $x$ increases by $1$ each step?
Which condition in a table confirms a linear rule when $x$ increases by $1$ each step?
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Constant first differences in $y$. Linear functions produce constant first differences in $y$-values for equal increments in $x$.
Constant first differences in $y$. Linear functions produce constant first differences in $y$-values for equal increments in $x$.
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Identify the rule from the table: $(x,y)$ are $(0,1)$, $(1,3)$, $(2,7)$, $(3,13)$. What is $y$?
Identify the rule from the table: $(x,y)$ are $(0,1)$, $(1,3)$, $(2,7)$, $(3,13)$. What is $y$?
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$y=x^2+x+1$. The table shows constant second differences of 2, indicating a quadratic with this form fitting all points.
$y=x^2+x+1$. The table shows constant second differences of 2, indicating a quadratic with this form fitting all points.
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Which condition in a table confirms a quadratic rule when $x$ increases by $1$ each step?
Which condition in a table confirms a quadratic rule when $x$ increases by $1$ each step?
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Constant second differences in $y$. Quadratic functions yield constant second differences in $y$-values for uniform $x$ increments.
Constant second differences in $y$. Quadratic functions yield constant second differences in $y$-values for uniform $x$ increments.
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Identify the rule from the table: $(x,y)$ are $(0,0)$, $(1,1)$, $(2,4)$, $(3,9)$. What is $y$?
Identify the rule from the table: $(x,y)$ are $(0,0)$, $(1,1)$, $(2,4)$, $(3,9)$. What is $y$?
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$y=x^2$. The points fit a quadratic pattern with $y$ as the square of $x$, showing constant second differences of 2.
$y=x^2$. The points fit a quadratic pattern with $y$ as the square of $x$, showing constant second differences of 2.
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Identify the rule from the table: $(x,y)$ are $(0,7)$, $(1,7)$, $(2,7)$, $(3,7)$. What is $y$?
Identify the rule from the table: $(x,y)$ are $(0,7)$, $(1,7)$, $(2,7)$, $(3,7)$. What is $y$?
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$y=7$. The constant y-values indicate a horizontal line, representing a constant function.
$y=7$. The constant y-values indicate a horizontal line, representing a constant function.
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Identify the rule: a graph is a parabola with vertex at $(0,0)$ passing through $(2,8)$. What is $y$?
Identify the rule: a graph is a parabola with vertex at $(0,0)$ passing through $(2,8)$. What is $y$?
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$y=2x^2$. The parabola's vertex and given point determine the coefficient 2 in the quadratic form.
$y=2x^2$. The parabola's vertex and given point determine the coefficient 2 in the quadratic form.
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Identify the rule: a graph is a line crossing the $y$-axis at $-1$ and rising $2$ for every run of $1$. What is $y$?
Identify the rule: a graph is a line crossing the $y$-axis at $-1$ and rising $2$ for every run of $1$. What is $y$?
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$y=2x-1$. The description gives slope 2 and y-intercept -1, directly forming the slope-intercept equation.
$y=2x-1$. The description gives slope 2 and y-intercept -1, directly forming the slope-intercept equation.
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Which condition in a graph confirms a relation is a function (vertical line test result)?
Which condition in a graph confirms a relation is a function (vertical line test result)?
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No vertical line intersects more than once. The vertical line test ensures each x-value maps to at most one y-value, defining a function.
No vertical line intersects more than once. The vertical line test ensures each x-value maps to at most one y-value, defining a function.
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Identify the rule: a graph is a line through $(0,4)$ and $(4,0)$. What is $y$ in terms of $x$?
Identify the rule: a graph is a line through $(0,4)$ and $(4,0)$. What is $y$ in terms of $x$?
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$y=-x+4$. The slope between the points is -1, with y-intercept 4, yielding this linear equation.
$y=-x+4$. The slope between the points is -1, with y-intercept 4, yielding this linear equation.
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