Some relationships cannot be described by polynomial functions. For example, let’s think about the relationship between the radius , in centimeters, and the surface area , in square centimeters, of the set of cylinders with a volume of 330 cm³ (this is a volume of 330 mL). What radius would result in the cylinder with the minimum surface area?

We know these formulas are true for all cylinders with radius , height , surface area , and volume :

Since we are only interested in cylinders with a volume of 330 cm³, we can use the volume formula to rewrite the surface area formula as:

Can you see how? We can use the volume formula rearranged as and then substitute for  in the formula for surface area.

We now have an equation giving as a function of for cylinders with a volume of 330 cm³. From the graph of shown here, we can quickly identify that a radius of about 3.75 cm results in a cylinder with minimum surface area and a volume of 330 cm³.​​​​​

is an example of a rational function. Rational functions are fractions with polynomials in the numerator and denominator. Polynomial functions are a type of rational function with 1 in the denominator.

Graph of a rational function with a point (3 point 46 comma 264 point 312), origin O. Horizontal axis labeled r, scale 0 to 8 by 2’s. Vertical axis labeled S, scale 0 to 600, by 200’s.

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In this situation, the height of a cylinder with fixed volume varies inversely with the square of the radius, , which means that, as the value of increases, the value of the height decreases, and vice versa. In later lessons, we’ll learn more about different features of rational functions, like why their graphs can look like they are made of two separate curves.