Problem 81 - College Algebra

Question

Describe how to graph a rational function.

Step-by-Step Solution

Answer

The graph of a rational function can be drawn by following these steps: 1) Identify the function. 2) Identify vertical asymptotes by setting the denominator equal to zero. 3) Identify horizontal asymptotes by taking the limit as x approaches infinity. 4) Find the x and y intercepts by setting y and x to zero respectively. 5) Plot the graph by marking the intercepts and asymptotes, and drawing the graph approaching but never crossing the asymptotes.

Step 1: Identify the function

The first step is to identify the function. Let's use a concrete example function \(f(x) = \frac{3x^2 - 2x + 1 }{x - 3}\).

Step 2: Identify vertical asymptotes

The next step is to find the vertical asymptotes of the rational function. This is done by setting the denominator of the function equal to zero and solving for \(x\). In this case, \(x - 3 = 0\), so \(x = 3\) is the vertical asymptote.

Step 3: Identify horizontal asymptotes

Next, identify horizontal asymptotes. This is done by considering the limit of the function as \(x\) approaches positive or negative infinity. As for our function, the degree of the numerator is equal to the degree of the denominator, so we use the ratio of the coefficients of the highest degree term in the numerator and denominator. That is, the horizontal asymptote is \(y = \frac{3}{1} = 3\).

Step 4: Find the X and Y intercepts

To find the \(x\)-intercepts, set \(y\) to zero and solve for \(x\). To find the \(y\)-intercept, set \(x\) to zero, and solve for \(y\). In our function, \(f(x) = 0\) leads to \(x = 0.5, -0.6667\) and \(f(0) = -0.3333\), so these are our X and Y intercepts respectively.

Step 5: Plotting the graph

The final step is to sketch the graph. Mark the asymptotes and intercepts on your graph. Then, draw the graph in regions divided by vertical asymptotes. Always remember that the graph should approach the asymptotes but never crosses them.