Variáveis dependentes e independentes

Roughly speaking, independent variables represent function inputs, while dependent variables represent function outputs. The value of a dependent variable depends on what the input is. However, the independent variable does not depend on anything; it is just whatever you want to input! We can examine the relationship between the dependent and independent variables.

But, before we discuss these concepts in detail, we need to know how to represent functions as formulas, which are special kinds of literal equations. That's actually how functions typically arise in word problems and real life: a relationship between an "input variable" (like number of hours worked) and an "output variable" (like amount of income earned), where the input determines the output.

Functions as Formula Equations

The most common way to write a function might be function notation $f(x)$, like in the function rule $f(x) = 3x+4$. The rule tells us that for any input $x$, the output "f of x" of the function $f$ will be $3x+4$. For example, $f(1) = 7$ while $f(10) = 34$.

Often, a function $f(x)$ is instead written as a formula $y = f(x)$, using one variable (often $x$) to represent the input and another (often $y$) to represent the output. For example, the function $f(x) = 3x+4$ can be expressed using the formula $y = 3x+4$. Then instead of saying $f(1) = 7$, we could say that $y=7$ when $x=1$. Or verbally, "the output is $7$ when the input is $1$".

Functions have one output for every input, but the function can be both linear and non-linear.

Note: Formulas are special literal equations that have one variable isolated in terms of the other variables. For example, $P = 4s$ is a formula for the perimeter of a square in terms of side length, while $s = \frac14 P$ is a formula for the side length of a square in terms of perimeter.

In literal equation form, functions have two types of numbers in them: constants and variables.

Constants are the numbers that stay the same in the function no matter what. In the equation $y = 3x + 4$, the number $4$ is a constant.

While 3 doesn't change, it is still attached to the variable. We call these coefficients.

Variables are "placeholders" for numbers that are often unspecified. In the equation form of a function, one variable (representing the output) will often depend on one or more other variables (representing the input).

For example, in the equation $y = 3x + 4$ from above, when the input value $1$ is substituted for $x$, the output value $y$ will be $7$, but when the input value $10$ is substituted for $x$, the output value $y$ will be $34$. The outputs will often (but not always) change as we vary the inputs.

In an equation describing a functional relationship, there are two different kinds of variables.

Independent Variables Represent Function Inputs

Independent variables are the ones we can freely manipulate, by plugging in any input value. In most math problems, we think of $x$ as the independent variable, since it's the one we usually change. We usually plot the independent variable on the horizontal axis of a graph (which we also call the $x$-axis).

Dependent Variables Represent Function Outputs

Dependent variables are determined by independent variables. (This is another way of saying, "a function spits out a single output when you plug in any input".) In most math problems, we think of $y$ as the dependent variable and plot it on the vertical axis (which we also call the $y$-axis).

Examples

Let's go back to the equation $y = 3x+4$. Plugging any input value for $x$ gives an output value for $y$. In other words, we could say the equation $y=3x+4$ makes the variable $y$ a function of the variable $x$. In this functional relationship between $x$ and $y$, the variable $x$ can be viewed as the independent variable while $y$ (whose value "depends" on the value of $x$) can be viewed as the dependent variable.

Tables & Graphs

Finally, we can also graph the equation $y = 3x+4$. For example, when the independent variable $x$ is $1$, the dependent variable takes the value $y=7$, so we plot the point $(1,7)$ (marked in green below). Notice how the horizontal $x$-axis represents the independent variable, while the vertical $y$-axis represents the dependent variable.

Besides graphing, you can also organize $(x,y)$ pairs into a table, like in the introduction to graphing.

Types of Variables

In math, a variable is a symbol (usually a letter) we use to represent a number whose value we don't know yet.

Two important types of variables in all types of math and science (shout-out to the science-lovers!) are independent variables (IV) and dependent variables (DV).

As with many math terms, there are several names for each of these types of variables that can be used interchangeably. Let's take a look.

Let's practice for a second. This might seem silly, but it's a good idea to make sure you're pretty comfortable with all of the possible terms for the dependent variable and independent variable.

So, what exactly are the IV and DV?

There are multiple ways to think about the IV and DV, and the exact definition of each type of variable often depends on the context you find them in.

For instance, in science, it's easiest to think about the IV and DV like this:

On the other hand, in math, it's often easier to think about the IV and DV like this:

To be clear, the dependent variable and independent variable serve the same functions in math and science. It's just the little details of how it's easiest to think about them that differ.

So, if one of the above definitions floats your boat more than the others, go with it! The math definitions will work in science and the science definitions will work in math.

One more little trick to identifying the dependent variable and independent variable—look at a graph!

Examples: Identifying Dependent and Independent Variables

Variables are values that are not known yet, or values that are subject to change. We typically deal with two kinds of variables: dependent and independent.

The two types of variables are related. Like the way they are named, the dependent variable changes as the independent variable changes. The dependent variable is the responding variable.

Dependent and independent variables are often used in scientific experiments. For example, if you want to see how the amount of water affects plant growth, you would consider the amount of water as the independent variable and the plant's height as the dependent variable. To get from an experiment to a test, you get a bunch of plants and give each plant a different amount of water. To conduct the experiment, you measure the effect of the water on the plant.

Some common independent variables are: time and amount of sunlight.

Let's take a look at some examples.

Identify the dependent and independent variable in each.

• Time of the year and temperature.

Time of the year is the independent variable and temperature is the dependent variable. The time of the year affects how cold or warm a day is.

• Dimensions of your room and length of decorative fairy lights you need.

The dimensions are the independent variable. If you have a larger room, you need more lights; if you have a smaller room, you need fewer lights.

• Monthly profit of a pizza shop and how many slices of pizza are sold.

Profit is the dependent variable. How much money the pizza shop makes is dependent on how many slices they sell.

Typically when we graph a function, we put the independent variable on the $x$-axis and the dependent variable on the $y$-axis.

(Video) Dependent vs Independent Variables (Simplifying Math)

by Eric Buffington

The video above explains how you can tell the difference between dependent and independent variables.

$\large{\text{Summary}}$

First, in order to have a dependent and independent variable, we must have an equation or function with at least $2$ variables where one variable impacts the other.

Function: A mathematical relationship in which the values of a dependent variable are uniquely determined by the values of one or more independent variables.

Dependent variable: The dependent variable follows the other variable, changes when other variables change, completely depend what happens to other variables.

Independent variable: The independent variable changes as it likes or changes independently, and forces the dependent variable to follow suit.

In order to have a dependent and independent variable, we must have an equation with at least $2$ variables where one variable impacts the other.

A great way to think about independent and dependent variables is to ask yourself the following question.

Which variable impacts the other one?

Example $1$:

While searching for golf balls, I can find $10$ for every hour that I search. The equation is $g=10h$ where $g$ is the number of golf balls I find and $h$ is the hours I spend looking.

"The number of golf balls I find depends on how many hours I spend looking."

• Independent variable: Hours spent looking for golf balls.
• Dependent variable: Number of golf balls I find.

Example $2$:

Each minute $(m)$ I spend studying improves my grade $(g)$ by $1$ point.

"My grade depends on how much I study."

• Independent variable: How much I study.
• Dependent variable: My grade.

You can also look at these problems in term of input and output. Input is independent, usually $x$. Output is dependent, usually $y$.