Source: ccarlstead, Flickr

The grade you make on a test depends on the number of questions you answered correctly. The independent variable is the number of questions you answered correctly and the dependent variable is the grade.

Source: bandita, Flickr

The number of calories I consume in a week determines the number of pounds I lose (or gain) that week. The number of calories I consume is the independent variable because the amount of weight lost (or gained) depends on the number of calories consumed.

Source: Joey Parsons, Flickr

A burning candle is an example of a functional relationship. The variables are the height of the candle and the amount of time the candle has been burning. The height of the candle is a function of the amount of time the candle has been burning. Since the height of the candle depends upon the amount of time the candle has been burning, the height of the candle is the dependent variable and the amount of time it has been burning is the independent variable.

Source: Cory Doctorow, Flickr

A vending machine can be an example of a functional relationship. The product dispensed from the machine is a function of the code entered or the button pressed. The variables are the code entered (or button pressed) and the product dispensed. The machine’s output (the product dispensed) depends on the input (the code entered or the button pressed). The input, or independent variable, causes a specific output, the dependent variable.

Source: NASA Glenn Research Center ImageNet

My heart rate is a function of the amount of activity I am engaged in. The variables are my heart rate and my activity level. The more strenuous the activity, the faster my heart beats. My heart rate depends on the level of the activity I am engaged in.

Functional Relationships in Symbolic Form

In the previous examples, we looked at real world situations and examined the functional relationship that exists. In Algebra, functional relationships may also exist within equations. Let’s take a look.

The equation Y = -5x – 4 is a functional relationship. The variables are x and y. The value of y depends upon the value of x. The independent variable is x and the dependent variable is y.

In other words, the answer, which in this case is y, depends on what is substituted for x. The answer y will change every time a new number is substituted for x. Therefore, y depends on the value of x.