A dependent variable is the output value that changes based on the input. It responds to changes in the independent variable: in y = 2x + 1, y is the dependent variable because its value depends on x. On a graph it goes on the vertical axis, because its value depends on what the input was.
In Spanish
variable dependiente
Also called
output variable, response variable, output
Grade level
6-8
Grade 6Grade 7Grade 8Algebra 1CCSS 6.EE.C.9SATACT
Dependent Variable in a picture
You pick the number of tickets; the cost follows from the rule. The cost is the dependent variable, so it goes on the vertical axis.
Formal definition and notation
When two quantities change in relation to each other, the quantity whose value is determined by the other is the dependent variable, and the quantity that can be chosen freely is the independent variable. An equation writes the dependent variable in terms of the independent one, as in C = 5n. For a functiony = f(x), y is the dependent variable and its possible values form the range.
Symbol
Read it as
Meaning
y
y
the usual letter for the dependent variable
f(x)
f of x
function notation for the output, the dependent variable, when the input is x
C = 5n
C equals five n
the dependent variable C written in terms of the independent variable n
Key facts
Ask ‘which one depends on the other?’ The answer to that question is the dependent variable.
In an equation solved for one letter, that letter is usually the dependent variable: in d = 60t, d depends on t.
In a table, the dependent variable is usually the second column or row; on a graph, it is the vertical axis.
In an experiment, the dependent variable is what you measure; the independent variable is what you change.
The same quantity can be dependent in one problem and independent in another, depending on the question.
Try it: feed the machine
Interactive
Type an input (the independent variable) and press Run. The output is the dependent variable: it depends on the input you chose.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Dependent Variable examples, step by step
Example 1: Pay for hours worked
Warm-up
A lifeguard earns P = 15h dollars for h hours. Which variable is dependent? Find P when h = 6.
The pay depends on the hours worked, so P is the dependent variable and h is independent.
Answer:P is the dependent variable. For 6 hours, the pay is 90 dollars.
Example 2: Write the equation
Core
A plant is 4 cm tall and grows 1.5 cm each day. Write an equation for its height H after d days, and find H after 10 days.
The height depends on the number of days, so H is dependent and d is independent.
Start at 4 and add 1.5 for each day: H = 4 + 1.5d.
For d = 10: H = 4 + 1.5 × 10 = 4 + 15 = 19.
Answer:H = 4 + 1.5d; after 10 days the plant is 19 cm tall.
Example 3: Variables in an experiment
Test-style
A student tests how grams of fertilizer f affect plant height H in centimeters and finds H = 20 + 3f. Name the dependent variable, then find H for 4 grams.
The student chooses the amount of fertilizer, so f is independent.
The height is measured as a result, so H is dependent.
H = 20 + 3 × 4 = 20 + 12 = 32.
Answer: The dependent variable is the plant height H; with 4 grams of fertilizer, H = 32 cm.
Common mistakes
✗ Mistake‘Hours studied depends on the test score,’ so hours is the dependent variable.
✓ InsteadThe test score depends on hours studied, so the score is dependent.
The dependent variable is usually the output, not the value you choose first. Read the relationship as ‘score depends on hours’.
✗ MistakeGraphing cost on the horizontal axis and number of tickets on the vertical axis.
✓ InsteadPut the independent variable (tickets) across and the dependent variable (cost) up.
By convention the dependent variable goes on the vertical axis, so a graph reads ‘as x grows, y does this’.
✗ MistakeIn d = 60t, t is dependent because it comes last.
✓ Insteadd is dependent: the distance is found from the time.
Look at which letter the equation is solved for and which value is found from the other, not at the order of the letters.
The independent variable is the input you choose or control; the dependent variable is the output that results. In C = 5n, n is independent and C is dependent.
An output is one particular value, like C = 15 for n = 3. The dependent variable is the quantity that takes all those output values.
Quick check: 3 questions
Question 1 of 3
In C = 5n, the cost C depends on the number of tickets n. Which variable is dependent?
Show answer and explanation
Answer: C.C is found from n, so C is dependent. n is the independent variable you choose, and 5 is a constant, not a variable.
Question 2 of 3
For y = 3x − 2, what is the value of the dependent variable when x = 4?
Show answer and explanation
Answer: A.y is dependent: y = 3 × 4 − 2 = 10. Choice B is the input itself, choice C adds 2 instead of subtracting, and choice D solves 3x − 2 = 4 for x.
Question 3 of 3
On a graph of distance traveled against time, where does distance belong?
Show answer and explanation
Answer: D. Distance depends on time, so distance is the dependent variable and goes on the vertical axis. Time, the independent variable, goes on the horizontal axis.
Dependent Variable: frequently asked questions
How do you identify the dependent variable?
Make a sentence with ‘depends on’: ‘The cost depends on the number of tickets’ sounds right, while ‘the number of tickets depends on the cost’ usually does not. The quantity before ‘depends on’ is the dependent variable. In an equation, it is usually the letter the equation is solved for, alone on one side, such as y in y = 4x + 3.
Is y always the dependent variable?
Usually, but not always. With x and y, y is dependent by convention. Real problems use letters that fit the story, such as d = 60t, where d is dependent. If you rewrite y = 2x as x = y2, then x becomes dependent. Decide from the relationship, not from the letter.
What is the difference between a dependent and an independent variable?
The independent variable is the input: you choose it or it changes on its own, like time or the number of items bought. The dependent variable is the output: it changes because the independent variable changed, like distance traveled or total cost. On a graph, independent goes across (horizontal) and dependent goes up (vertical).
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.EE.C.9
Cite this entry
Effortless Math Team. “Dependent Variable.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/dependent-variable/
Flashcard
TermDependent Variablethe output that depends on the input
DefinitionThe variable whose value depends on another. In C = 5n, cost C depends on tickets n; it goes on the vertical axis.
RelationA relation is any set of ordered pairs that links inputs to outputs, such as {(1, 4), (2, 5), (2, 7)}. It can be shown as a list, table, mapping diagram or graph. A relation in which every input has exactly one output is a function; this one is not, because 2 has two outputs.Functions · Grade 8, Algebra 1Independent VariableAn independent variable is an input value that can be chosen or changed. The dependent variable changes based on it. In y = 3x, x is the independent variable and y is the dependent one. On a graph the independent variable goes on the horizontal axis; in an experiment it is the factor you change on purpose.Functions · Grade 6, Grade 7, Grade 8FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2Function NotationFunction notation is the way of writing a function as f(x), read “f of x”: f names the function, x is the input, and f(x) is the output. Writing f(x) = 2x + 5 means the same as y = 2x + 5, and f(3) = 11 says that the input 3 gives the output 11.Functions · Grade 8, Algebra 1, Algebra 2SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2GramA gram is a metric unit of mass, about the mass of a paper clip. Its symbol is g. There are 1,000 grams in a kilogram and 1,000 milligrams in a gram. Grams measure light things, such as food portions, coins and letters; heavier things are measured in kilograms.Measurement · Grade 3, Grade 4, Grade 5CentimeterA centimeter is a metric unit of length equal to one hundredth of a meter, about the width of a fingernail. Its symbol is cm. There are 100 centimeters in 1 meter and 10 millimeters in 1 centimeter, and 1 inch equals 2.54 centimeters.Measurement · Grade 3, Grade 4, Grade 5OutputAn output is the value a function or rule produces from a given input. You put a number in, apply the rule, and the result that comes out is the output. For y = 2x, the input 3 gives the output 6. In an input-output table the output sits in the second column.Functions · Grade 6, Grade 7, Grade 8SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8Domain and RangeDomain and range describe a function: the domain is the set of all inputs it accepts, and the range is the set of all outputs it produces. For the points (1, 4) and (2, 5), the domain is {1, 2} and the range is {4, 5}. Restrictions usually come from division by zero or square roots of negatives.Functions · Grade 8, Algebra 1, Algebra 2GraphA graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Functions · Grade 5, Grade 6, Grade 7Mathematical ModelA mathematical model represents a real situation using numbers, equations, diagrams, or graphs, well enough to make predictions. The equation C = 15 + 9m can model the cost of a subscription after m months. Every model simplifies, and knowing what a model leaves out is as important as knowing what it includes.Problem Solving · Grade 6, Grade 7, Grade 8InputAn input is the value you put into a function or a rule. The rule acts on the input to produce the output: for f(x) = x + 2, the input 5 gives the output 7. For many functions, the input is the x-value. In an input-output table, the inputs fill the first column.Functions · Grade 6, Grade 7, Grade 8