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This printable supports Common Core Mathematics Standard HSF-BF.A.1, HSF-BF.A.1a

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Writing Function Rules From Tables (Grades 11-12)

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Writing Function Rules From Tables

1.
Which of the function rules describes the data in the table below?

 $\ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ -2 \ \ \ \ \ \$ $\ \ \ \ \ \ -1 \ \ \ \ \ \$ $\ \ \ \ \ \ 0 \ \ \ \ \ \$ $\ \ \ \ \ \ 1 \ \ \ \ \ \$ $\ \ \ \ \ \ 2 \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \$ $8$ $1$ $0$ $5$ $16$
1. $f(x) = -x + 6$
2. $f(x) = 3x^2 + 2x$
3. $f(x) = 5x^2 + 6x$
4. $f(x) = 134 e^(0.0036x)$
2.
Determine the function rule which models the data from the table.

 $\ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ -2 \ \ \ \ \ \$ $\ \ \ \ \ \ -1 \ \ \ \ \ \$ $\ \ \ \ \ \ 0 \ \ \ \ \ \$ $\ \ \ \ \ \ 1 \ \ \ \ \ \$ $\ \ \ \ \ \ 2 \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \$ $-45$ $-25$ $-5$ $15$ $35$
1. $f(x) = 20x - 5$
2. $f(x) = -1/2 x^2 + 37.2 x - 6$
3. $f(x) = -20x - 5$
4. $f(x) = 20x$
3.
Which function correctly describes the data found in the following table?

 $\ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ -2 \ \ \ \ \ \$ $\ \ \ \ \ \ -1 \ \ \ \ \ \$ $\ \ \ \ \ \ 0 \ \ \ \ \ \$ $\ \ \ \ \ \ 1 \ \ \ \ \ \$ $\ \ \ \ \ \ 2 \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \$ $29/4$ $6$ $21/4$ $5$ $21/4$
1. $f(x) = 1/8 x^2 - 5/8 x + 21/4$
2. $f(x) = 3/8 x^2 - 3/8 x + 21/4$
3. $f(x) = 1/8 x^2 - 7/8x + 21/4$
4. $f(x) = 1/4 x^2 - 1/2 x + 21/4$
4.
Given the values in the table below, choose the function which best describes them. The values in the table may vary sightly from the output of the function rule, due to rounding.

 $\ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ -2 \ \ \ \ \ \$ $\ \ \ \ \ \ -1 \ \ \ \ \ \$ $\ \ \ \ \ \ 0 \ \ \ \ \ \$ $\ \ \ \ \ \ 1 \ \ \ \ \ \$ $\ \ \ \ \ \ 2 \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \$ $0.02$ $0.07$ $0.18$ $0.50$ $1.36$
1. $f(x) = 0.05x + 0.18$
2. $f(x) = 0.5 e^(x-1)$
3. $f(x) = 0.1 e^(0.69x)$
4. $f(x) = 0.2 e^(1.1x)$
5.
The values in the table come from a function, either linear, quadratic, or exponential. If it is an exponential function, assume that it is of the form $f(x) = a b^(x)$, where $a, b in RR, b!=0,1$.

 $\ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ -2 \ \ \ \ \ \$ $\ \ \ \ \ \ -1 \ \ \ \ \ \$ $\ \ \ \ \ \ 0 \ \ \ \ \ \$ $\ \ \ \ \ \ 1 \ \ \ \ \ \$ $\ \ \ \ \ \ 2 \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \$ $1.25$ $2.5$ $5$ $10$ $20$
A.
Using the differences of y-values, or the differences of the differences, determine whether the function values in the table are from a linear, quadratic, or exponential function.
1. Linear
3. Exponential
B.
Given the answer in the previous question, how many points are needed to determine the function rule that describes the values?
1. 2
2. 3
3. 4
4. All of them.
C.
Find the function rule that describes the data from the table.

6.
The values in the table below come from a quadratic function. Determine the algebraic function rule.

 $\ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ -5 \ \ \ \ \ \$ $\ \ \ \ \ \ -4 \ \ \ \ \ \$ $\ \ \ \ \ \ -3 \ \ \ \ \ \$ $\ \ \ \ \ \ -2 \ \ \ \ \ \$ $\ \ \ \ \ \ -1 \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \$ $120$ $85$ $56$ $33$ $16$

7.
The values in the table come from an exponential function of the form $f(x) = a * e^(bx)$, where $a, b in RR$. The function values in the table have been rounded to two decimal places. Find the numeric values of the coefficients $a$ and $b$.

 $\ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ 0 \ \ \ \ \ \$ $\ \ \ \ \ \ 1 \ \ \ \ \ \$ $\ \ \ \ \ \ 2 \ \ \ \ \ \$ $\ \ \ \ \ \ 3 \ \ \ \ \ \$ $\ \ \ \ \ \ 4 \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \$ $-3$ $-1.82$ $-1.10$ $-0.67$ $-0.41$

8.
The values in the table come from either a linear, a quadratic, or an exponential function. If it is an exponential function, assume that it takes the form $f(x) = a*e^(b x)$, where $a, b in RR$. Determine the function rule that describes the data in the table. Round answers to two decimal places if necessary.

 $\ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ 10 \ \ \ \ \ \$ $\ \ \ \ \ \ 11 \ \ \ \ \ \$ $\ \ \ \ \ \ 12 \ \ \ \ \ \$ $\ \ \ \ \ \ 13 \ \ \ \ \ \$ $\ \ \ \ \ \ 14 \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \$ $-5.28$ $-5.84$ $-6.4$ $-6.96$ $-7.52$

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