Function Rules From Data - ISEE Upper Level Mathematics Achievement

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

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

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

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

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

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

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

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

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 $y$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    $m=\Delta y$ 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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