ISEE Upper Level Mathematics Achievement Flashcards: Function Rules From Data

What this deck covers

This deck focuses on Function Rules From Data, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Upper Level Mathematics Achievement.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Identify the rule from the table: $(x,y)$ are $(0,1)$, $(1,3)$, $(2,7)$, $(3,13)$. What is $y$?

Answer: $y=x^2+x+1$. The table shows constant second differences of 2, indicating a quadratic with this form fitting all points.

Flashcard 2: Identify the rule from the table: $(x,y)$ are $(0,5)$, $(1,2)$, $(2,-1)$, $(3,-4)$. What is $y$?

Answer: $y=-3x+5$. The table shows constant first differences of -3 in $y$, indicating a slope of -3 and $y$-intercept of 5.

Flashcard 3: Identify the rule from the table: $(x,y)$ are $(-2,4)$, $(-1,1)$, $(0,0)$, $(1,1)$, $(2,4)$. What is $y$?

Answer: $y=x^2$. This symmetric set of points around the vertex at (0,0) matches the standard quadratic form.

Flashcard 4: Identify the rule: a graph is a line through $(0,4)$ and $(4,0)$. What is $y$ in terms of $x$?

Answer: $y=-x+4$. The slope between the points is -1, with y-intercept 4, yielding this linear equation.

Flashcard 5: Identify the rule: a graph is a line crossing the $y$-axis at $-1$ and rising $2$ for every run of $1$. What is $y$?

Answer: $y=2x-1$. The description gives slope 2 and y-intercept -1, directly forming the slope-intercept equation.

Flashcard 6: Which condition in a table confirms a quadratic rule when $x$ increases by $1$ each step?

Answer: Constant second differences in $y$. Quadratic functions yield constant second differences in $y$-values for uniform $x$ increments.

Flashcard 7: Identify the rule: points $(0,3)$ and $(2,7)$ lie on a line. What is $y$ in terms of $x$?

Answer: $y=2x+3$. The slope calculated from the points is 2, and the $y$-intercept is 3, yielding the linear equation.

Flashcard 8: Identify the rule from the table: $(x,y)$ are $(0,7)$, $(1,7)$, $(2,7)$, $(3,7)$. What is $y$?

Answer: $y=7$. The constant y-values indicate a horizontal line, representing a constant function.

Flashcard 9: Identify the rule from the table: $(x,y)$ are $(2,10)$, $(4,20)$, $(6,30)$. What is $y$?

Answer: $y=5x$. The points show direct proportionality with a constant ratio of 5, yielding this linear rule through the origin.

Flashcard 10: Which condition in a table confirms a linear rule when $x$ increases by $1$ each step?

Answer: Constant first differences in $y$. Linear functions produce constant first differences in $y$-values for equal increments in $x$.

Flashcard 11: What is the point-slope form of a line with slope $m$ passing through $(x_1,y_1)$?

Answer: $y-y_1=m(x-x_1)$. This form derives from the slope formula, expressing the line using a known point and slope.

Flashcard 12: Identify the rule: a line passes through $(3,-2)$ and has slope $5$. What is $y$ in terms of $x$?

Answer: $y=5x-17$. Using point-slope form with the given point and slope yields this equation after simplification.

Flashcard 13: Which condition in a graph confirms a relation is a function (vertical line test result)?

Answer: 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.

Flashcard 14: Which table pattern indicates an inverse variation rule $y=\frac{k}{x}$?

Answer: Constant product $xy=k$. Inverse variation produces a constant product of x and y across the table entries.

Flashcard 15: Identify the rule from the table: $(x,y)$ are $(1,3)$, $(2,6)$, $(3,12)$, $(4,24)$. What is $y$?

Answer: $y=3\cdot 2^{x-1}$. The y-values double each time x increases by 1, fitting an exponential model with base 2 adjusted for the initial value.

Flashcard 16: Identify the rule from the table: $(x,y)$ are $(0,0)$, $(1,1)$, $(2,4)$, $(3,9)$. What is $y$?

Answer: $y=x^2$. The points fit a quadratic pattern with $y$ as the square of $x$, showing constant second differences of 2.

Flashcard 17: Identify the rule: a line has slope $-4$ and $y$-intercept $6$. What is the equation?

Answer: $y=-4x+6$. The slope-intercept form directly incorporates the given slope and $y$-intercept into the linear equation.

Flashcard 18: What is the general rule for a linear function written in slope-intercept form?

Answer: $y=mx+b$. This form expresses a linear function with slope $m$ and $y$-intercept $b$.

Flashcard 19: Identify the rule from the table: $(x,y)$ are $(1,4)$, $(2,7)$, $(3,10)$, $(4,13)$. What is $y$?

Answer: $y=3x+1$. The table exhibits constant first differences of 3 in $y$, confirming a linear rule with slope 3 and fitting the points.

Flashcard 20: Identify the rule from the table: $(x,y)$ are $(0,2)$, $(1,6)$, $(2,18)$, $(3,54)$. What is $y$?

Answer: $y=2\cdot 3^x$. The y-values multiply by 3 each time x increases by 1, matching an exponential with base 3 and initial factor 2.

Flashcard 21: What is the slope formula for a line through points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. 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.

Flashcard 22: Which table pattern indicates an exponential rule when $x$ increases by $1$ each step?

Answer: Constant ratio $\frac{y_{k+1}}{y_k}$. Exponential functions maintain a constant ratio between consecutive y-values for equal x increments.

Flashcard 23: Identify the rule from the table: $(x,y)$ are $(1,8)$, $(2,4)$, $(4,2)$, $(8,1)$. What is $y$?

Answer: $y=\frac{8}{x}$. The product $xy=8$ is constant, indicating inverse variation with this form fitting all points.

Flashcard 24: Identify the rule: a graph is a parabola with vertex at $(0,0)$ passing through $(2,8)$. What is $y$?

Answer: $y=2x^2$. The parabola's vertex and given point determine the coefficient 2 in the quadratic form.

Flashcard 25: What is slope in a table when $x$ increases by $1$ each row (in terms of $\Delta y$)?

Answer: $m=\Delta y$ per $1$ unit of $x$. When $x$ increases by 1 unit per row, the slope equals the constant change in $y$ for each step.