Using Tables to Find Function Rules

4.4.3 Using Tables to Find Function Rules

4.4.3 • Using Tables to Find Function Rules

Activity

1. A square that has a side length of 9 cm has an area of 81 $c m^{2}$ . The relationship between the side length and the area of the square is a function.

For questions a-e use the table by answering each question.

Side Length $(c m)$	Area $(c m^{2})$
1	a. _______
2	b. _______
4	c. _______
6	d. _______
$s$	e. _______

This table lists side lengths in centimeters and requires calculation of the corresponding areas in square centimeters, likely for a square or similar polygon.

a. What is the area in $c m^{2}$ when the side length is $1cm$ ?

The area is 1 $c m^{2}$ when the side length is $1cm$ .

b. What is the area in $c m^{2}$ when the side length is $2cm$ ?

The area is 4 $c m^{2}$ when the side length is $2cm$ .

c. What is the area in $c m^{2}$ when the side length is $4cm$ ?

The area is ${16cm}^{2}$ when the side length is $4cm$ .

d. What is the area in $c m^{2}$ when the side length is $6cm$ ?

The area is ${36cm}^{2}$ when the side length is $6cm$ .

e. Write a rule for a function, $A$ , that gives the area of the square in $c m^{2}$ when the side length is $s$ $c m$ .

Compare your answer:

$A (s) = s^{2}$

f. What does $A (2)$ represent in this situation? What is its value? Write your answers, then click solution to compare your answer.

Compare your answer:

$A (2)$ is the area of the square in cm2 when the side length is 2 cm. Its value is 4.

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g. Sketch a graph of this function. Use the desmos graphing tool or technology outside the course.

Compare your answer:

A graph showing area (cm²) on the y-axis and side length (cm) on the x-axis. The curve rises steeply, illustrating that as side length increases, area increases rapidly. Points are marked at side lengths 1, 2, 4, and 6 cm.

2. A roll of paper that is 3 feet wide can be cut to any length.

Roll of paper, slightly unrolled. Its width is 3 feet.

a. If we cut a length of 2.5 feet, what is the perimeter of the paper? Enter your answer in feet.Use integers only.

The perimeter of the paper is 11 feet.

For questions b-f, keep in mind that the roll of paper is 3 feet wide. Complete the table by answering each question.

side length (feet)	perimeter (feet)
1	b. _______
2	c. _______
6.3	d. _______
11	f. _______

b. If we cut a length of 1 foot, what is the perimeter of the paper to the nearest whole foot?

The perimeter is 8 feet when the side length is $1cm$ $(1 + 3 + 1 + 3 = 8)$ .

c. To the nearest whole foot, what is the perimeter when the side length is 2 feet?

The perimeter is 10 feet when the side length is $2cm$ $(2 + 3 + 2 + 3 = 10)$ .

d. What is the exact perimeter, in feet, when the side length is 6.3 feet?

The perimeter is 18.6 feet when the side length is $6.3cm$ $(6.3 + 3 + 6.3 + 3 = 18.6)$ .

e. What is the perimeter, to the nearest whole foot,when the side length is 11 feet?

The perimeter is 28 feet when the side length is $11cm$ $(11 + 3 + 11 + 3 = 28)$ .

f. Write a rule for a function, P, that gives the perimeter of the paper in feet when the side length in feet is l.

Compare your answer:

$P (l) = 6 + 2 l$

g. What does P(11) represent in this situation? What is its value?

Compare your answer:

$P (11)$ is the perimeter of the paper in feet when the paper is cut at a length of 11 feet. Its value is 28.

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h. Sketch a graph of this function. Use the desmos graphing tool or technology outside the course.

Compare your answer:

A line graph showing the relationship between side length (feet) on the x-axis and perimeter (feet) on the y-axis. The line passes through (1, 8), (2, 10), and (6.3, 18.6), indicating a linear increase.

Video: Writing Function Rules

Watch the following video to learn more about writing rules with function notation.

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Self Check

Which function rule can describe the table below?

$x$	$y$
1	3
2	5
5	11

$f(x)=x+2$
$f(x)=2x+2$
$f(x)=2x+1$
$f(x)=x+3$

Additional Resources

Write the Function Rule from a Table

Kelly can tune 6 cars in 3 hours. The table below is completed to show this relationship. The last column has the output for any $x$ . This is the rule of the function.

Time spent tuning cars ( $x$ )	2	3	6	7	$x$
Number of cars tuned up ( $y$ )	4	6	12	14	$2 x$

The table demonstrates the relationship between time spent tuning cars (x) and the number of cars tuned up (y), illustrating the formula y=2x through numerical examples.

Kelly can tune 6 cars in 3 hours or 2 cars in 1 hour, so the function rule is $f (x) = 2 x$ .

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Write the Function Rule from a Table

Determine the rule in function notation for the table below:

$x$ (input)	1	2	3	4	$x$
$y$ (output)	5	8	11	14	?

This table presents input (x) and corresponding output (y) value pairs, demonstrating a functional relationship, with a general 'x' input and an unknown 'y' output.

Compare your answers:

Here is how to find the rule in function notation:

In the table, every $x$ is multiplied by 3, and then 2 is added. The rule is:

$f (x) = 3 x + 2$