Creating Tables from Verbal Descriptions

1.13.2 Creating Tables from Verbal Descriptions

1.13.2 • Creating Tables from Verbal Descriptions

Activity

Creating a Table of Values

For each situation in numbers 1 – 3, create a table of values with at least five entries to represent the situation.

Julio works at a clothing store and makes $14 per hour.

Compare your answer:

Number of hours	0	1	2	3	4	5
Amount earned	0	14	28	42	56	70

Earnings based on hours worked, showing a direct relationship of $14 earned per hour.

Jake had cavities at his last dentist visit. The dentist told him to brush his teeth for two minutes each time he brushes.

Compare your answer:

Amount earned	1	2	3	4	5
Total number of minutes brushed	2	4	6	8	10

This table displays corresponding values for amount earned and total minutes brushed, showing a direct 1:2 proportional relationship.

For a fundraiser, Celia ran 5 miles on Monday (day 0) and then 3 miles every day after that.

Compare your answer:

$x$	0	1	2	3	4	5
$y$	5	8	11	14	17	20

Pairs of x and y values for the linear function y = 3x + 5.

Using a Table of Values to Write Equations

Tables of values can be used to write equations. The tables from Creating a Table of Values all represent linear equations.

For questions 4 – 8, revisit the table from question 1 and write an equation to represent the table of values. Notice that columns for the $x$ and $y$ values are added.

$x$	0	1	2	3	4	5
$y$	0	14	28	42	56	70

This table displays corresponding values for variables x and y, illustrating their linear relationship where y = 14x.

What is the starting value (when $x = 0$ , what does $y$ equal)?

$y = 0$

How would you describe this value on the graph?

Compare your answer:

$y$ -intercept

What is the rate of change, or the change between the $y$ -values over the change between the $x$ -values?

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Compare your answer:

$\frac{14}{1}$ or 14

How would you describe this value on the graph?

Compare your answer:

slope

Using the information in questions 4 – 7, what would the equation be for the values represented by the table? Use slope-intercept form.

Compare your answer:

$y = 14 x + 0$ or $y = 14 x$

Write an equation from the table of values created in question 2 of Creating a Table of Values.

Hint: Work backward to find the value where $x = 0$ using the pattern set in the table of values.

$x$	0	1	2	3	4	5
$y$	?	2	4	6	8	10

Table showing corresponding x and y values for a linear function, y=2x, with an unknown y-value at x=0.

Compare your answer:

$y = 2 x$ (note: subtract the top row by 1 and the bottom row by 2 to go back to where $x = 0$ for the $y$ -intercept).

10.

Write an equation from the table of values created in question 3 of Creating a Table of Values

$x$	0	1	2	3	4	5
$y$	?	2	4	6	8	10

This table presents corresponding values for variables x and y, illustrating a potential linear relationship where y=2x for x>=1 and an initial unknown y-value.

Compare your answer:

$y = 3 x + 5$

Are you ready for more?

Extending Your Thinking

Kylie pays $20 to rent a scooter plus $0.40 per mile she drives the scooter. Create a table of values to represent Kylie’s situation starting with $x = 0$ .

Compare your answer:

0	1	2	3	4	5
20	20.40	20.80	21.20	21.60	22.10

Use the table to determine how much she has to pay after 5 miles.

Compare your answer:

Kylie would pay $22.10.

Video: Writing an Equation from a Table of Values

Watch the following video to learn more about writing equations from a table of values.

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Create Your Own Situation

Think of your own situation that could be modeled by a linear equation. Create a table of values that represents that situation. Write your problem down then switch with a partner. Make a table of values for each other’s problems. Then, check your work together.

Next, plot the points of each table of values in a coordinate plane. Answer the questions after discussing with your partner:

Describe the scenario created by your partner.

Compare your answer:

Answers vary.

Use Desmos to graph the table of values to represent your partner’s scenario.

Use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the Desmos tool below.

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Compare your answer:

Answers vary.

Does your partner’s data model a linear relationship or not? Describe how you determined your answer.

Compare your answer:

The points form a line. This happens because the same number was added to each value in the top row and the same number was added to each value in the bottom row of the tables.

Self Check

Which of the following equations could be represented by this table of values?

$x$	0	1	2	3
$y$	20	23	26	29

$y = 20x + 3$
$y = 20x$
$y = \frac{1}{3}x + 20$
$y = 3x +20$

Additional Resources

Writing Linear Equations from Tables

When writing linear equations from tables, it is important to identify the slope and the $y$ -intercepts from the table.

Example

Given the table, write an equation in slope-intercept form.

$x$	0	1	2	3	4
$y$	12	16	20	24	28

This table presents five (x,y) data pairs, illustrating a linear relationship where y increases by 4 for each unit increase in x.

Step 1 - Identify the $y$ -intercept.

The $y$ -intercept is the starting value, when $x = 0$ .

Looking at the table, when $x = 0$ , the $y$ -value is 12, so this is the $y$ -intercept.

Step 2 - Identify the slope.

Slope is the change in $y$ over the change in $x$ .

The $y$ -values change by 4 each time. The $x$ -values change by 1 each time.

The slope is $\frac{4}{1}$ or 4.

Step 3 - Write the equation in slope-intercept form.

Slope intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the $y$ -intercept.

Substitute the values, $y = 4 x + 12$ .

Try it

Writing Linear Equations from Tables

Given the table, write an equation in slope-intercept form.

$x$	0	1	2	3
$y$	6	11	16	21

Table presenting four pairs of (x, y) values that demonstrate a linear relationship, specifically y = 5x + 6.

Compare your answer:

Here is how to write the equation in slope-intercept form:

Step 1 - Identify the $y$ -intercept.

The $y$ -intercept is the starting value, when $x = 0$ .

Looking at the table, when $x = 0$ , the $y$ -value is 6, so this is the $y$ -intercept.

Step 2 - Identify the slope.

Slope is the change in $y$ over the change in $x$ .

The $y$ -values change by 5 each time.

The $x$ -values change by 1 each time. The slope is $\frac{5}{1}$ or 5.

Step 3 - Write the equation in slope-intercept form.

Slope intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the $y$ -intercept.

Substitute the values, $y = 5 x + 6$ .