Graphing Rational Functions

Graphing a rational function reveals how outputs change with inputs, highlighting key features like domain, range, and overall behavior.

Imagine designing a 500 cubic centimeters aluminum can where top and bottom material costs 0.04 cents per square centimeter and side material costs 0.02 cents per square centimeter.

Using these material costs and the fixed volume, the total cost is modeled by the rational function C(r), where r is the radius in centimeters.

Because a physical radius must be strictly positive, the domain of r is zero to infinity.

As we plot the graph as r approaches zero, the term 20 over r becomes extremely large, driving the cost sharply upward and creating a vertical asymptote at r equals 0.

As the radius grows very large, the r-squared term dominates, sending cost back towards infinity.

Because cost rises at both ends, plotting the function forms a smooth, U-shaped curve which dips down to a lowest point in between.

Using a graphing calculator to trace this dip gives the minimum point (3.41, 8.79). So, our range includes all values greater than or equal to about 8.79 cents.