Question

f(x)=2x^2+16/x

Answer 1

STEP 1:

Factor the numerator and denominator of R and find its domain. If 0 is in the domain, find the y-intercept, R(0), and plot it.

STEP 2:

Write R in lowest terms as p(x)q(x) and find the real zeros of the numerator; that is, find the real solutions of the equation p(x) = 0, if any. These are the x-intercepts of the graph. Determine the behavior of the graph of R near each x-intercept, using the same procedure as for polynomial functions. Plot each x-intercept and indicate the behavior of the graph near it.

STEP 3:

With R written in lowest terms as p(x)q(x), find the real zeros of the denominator; that is, find the real solutions of the equation q(x) = 0, if any. These determine the vertical asymptotes of the graph. Graph each vertical asymptote using a dashed line.

STEP 4:

Locate any horizontal or oblique asymptotes using the procedure given in the previous section. Graph the asymptotes using a dashed line. Determine the points, if any, at which the graph of R intersects these asymptotes. Plot any such points.

STEP 5:

Using the real zeros of the numerator and the denominator of the given equation for R, divide the x-axis into intervals and determine where the graph is above the x-axis and where it is below the x-axis by choosing a number in each interval and evaluating R there. Plot the points found.

STEP 6:

Analyze the behavior of the graph of R near each asymptote and indicate this behavior on the graph.

STEP 7:

Put all the information together to obtain the graph of R