Independent & Dependent Variables in Tables and Graphs

The independent variable is the input; the dependent variable is the output determined by that input in the relationship. On a typical input-output graph, the independent variable goes on the horizontal x-axis and the dependent variable on the vertical y-axis. In a table, read the headings and context first: input values are often on the left, but columns can be reversed.

How to identify independent and dependent variables

Ask: “Which quantity are we using to determine the other?” If wages are calculated from hours worked at a fixed rate, hours are the input and wages are the output. In the equation y = 3x, choose a value of x and multiply it by 3 to find y.

• Independent: hours worked, x.
• Dependent: wages earned in dollars, y.
• Relationship: y = 3x, assuming a $3 hourly rate with no other pay.

Independent does not mean “unaffected by anything in the world.” It describes the variable’s role in the model. Time, for example, can be the input even though you cannot control time. An observed association between two variables also does not, by itself, prove cause and effect.

Finding variables in a table

Read the labels before choosing a column. For wages calculated from hours, hours remain independent even if wages are printed first:

For the row with 2 hours, the output is $6. Write the ordered pair as (2, 6), with input first, even though that input is in the table’s right column. The equation is y = 3x. A pattern alone may fit several rules, so use the stated relationship as well as the numbers.

Finding variables on a graph

For a graph of y as a function of x, the usual convention places input x horizontally and output y vertically. Always read the axis labels and the question. The names of the quantities matter more than simply calling every horizontal quantity a cause.

To read the point (3, 9), start at 3 hours on the horizontal axis, move up to the plotted point, then read $9 on the vertical axis. Hours are independent because the model uses hours to calculate wages; wages are dependent. The table, plotted points and equation all describe the same relationship.

A dependent variable need not increase when the input increases. For example, remaining water may decrease as time passes. Relationships can also be nonlinear: the area A of a square depends on its side length s through A = s².

Worked example: a starting height plus growth

A plant is 3 cm tall at the start and grows 5 cm per week in a simple constant-growth model. Let x be weeks since the start and y be height in centimeters.

Independent: weeks x. Dependent: height y. Equation: y = 5x + 3.

At week 1, y = 5 × 1 + 3 = 8 cm. The 3 is the height at week 0. If a problem instead says the plant is 3 cm tall after one week, that is a different starting condition; y = 5x + 3 would not fit that statement.

Practice: identify the variables and check your answers

Try each question before opening its answer.

1. A taxi charges $3 per mile with no starting fee. Name the variables and write the equation.
   Show answer 1

   Miles traveled x are independent; fare y is dependent. The model is y = 3x.

2. A table lists x values 2, 4, 6, 8 and y values 10, 20, 30, 40. It states that x is the input. Identify the output and the rule y = kx.
   Show answer 2

   The dependent output is y. Since y ÷ x = 5 in each row, k = 5 and y = 5x.

3. A graph uses temperature horizontally and ice creams sold vertically to model sales from temperature. Which variable is independent? Does the graph alone prove causation?
   Show answer 3

   Temperature is the independent input in this model; sales are the dependent output. The graph alone does not establish cause and effect.

4. Find the output values for y = 6x − 1 when x = 0, 1, 2, 3.
   Show answer 4

   The outputs are −1, 5, 11, 17, respectively. For example, x = 2 gives y = 12 − 1 = 11.

5. A store receives $12 for each item sold. Name the variables and write revenue y in terms of items x.
   Show answer 5

   Items sold x are independent; revenue y is dependent. The equation is y = 12x. Revenue is not the same as profit.

6. A runner has already burned 50 calories and burns 8 more per minute in a model. Write total calories y after x additional minutes.
   Show answer 6

   The model is y = 8x + 50. Additional minutes x are independent; total calories y are dependent.

For more practice, use the free Algebra 1 worksheets and Algebra 1 Online Center. Continue with completing a table and graphing a two-variable equation when you are ready to connect all three representations.

Optional calculator and video lessons

After you identify the input, the function calculator can help check substitution into an equation. It calculates outputs; it does not decide which quantity is independent in a word problem.

The existing videos provide another explanation and follow-up graphing practice.

Independent and dependent variables

Two-variable equations and graphs

Learning sources

Common Core 6.EE.C.9 connects dependent and independent quantities through equations, tables and graphs. For the distinction between association and causation, see OpenStax’s discussion of scatter plots and correlation. The examples and exercises above are original teaching examples.

Related Topics

• Complete a Table and Graph a Two-Variable Equation
• Using a Table to Write Down a Two Variable Equation
• Objects On a Coordinate Plane
• Understanding Quadrants
• Ratio Tables

Where each variable goes: graph and table

Tables do not always follow the usual layout, so read the headings instead of trusting the position.

Three questions to tell them apart

1. Which value do I choose or control? That is the independent variable. Time, hours, number of items and amount of water are common examples.
2. Which value depends on the other? That is the dependent variable. Ask: does this change because the other one changed?
3. Does the sentence still make sense with ‘depends on’? ‘Total cost depends on the number of tickets’ makes sense. ‘The number of tickets depends on total cost’ does not, so tickets is independent.

Try it: a taxi charges $3 plus $2 per mile. The fare depends on the miles, so miles go on the x-axis and the fare goes on the y-axis. The equation is y = 2x + 3, and a 5-mile ride costs 2(5) + 3 = $13.

Frequently Asked Questions

Which one is the independent variable in a graph?

The independent variable is the one on the horizontal x-axis. It is the input, the value that is chosen or changes on its own, such as time or hours worked. The dependent variable is plotted on the vertical y-axis because its value depends on the input.

Where do dependent variables go on a graph?

Dependent variables go on the vertical y-axis. As you move right along the x-axis, the height of each point shows the dependent value for that input. For example, if a plant grows 5 cm each week from 3 cm, the height in centimeters goes on the y-axis.

Is the independent variable on the left or right of a table?

It is usually on the left. When a table runs across in rows, it is usually the top row. This layout is a habit, not a rule, so always read the column headings. Whatever the table lists as input, like weeks or minutes, is the independent variable.

Is time always the independent variable?

Time is almost always the independent variable, because it moves forward no matter what else happens. If a graph shows distance traveled over time or money saved each month, time goes on the x-axis. Exceptions are rare, such as when you ask how long a task takes for different amounts of work.