Using the 7 steps outlined in section 4.3 of your book, analyze the graph of the following function:
R(x) = x^2 - 49/x^4 - 16
Step 1: Factor the numerator and denominator of R and find its domain. If 0 is in the domain, find 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, fnd 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 intercepts 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.
3 answers
1. (x-7)(x+7) / (x-2)(x+2)(x^2+4)
2. zeros: x=7,-7
3. vert asymp: x = 2,-2
4. degree of top < degree of bottom: hor. asymp at y=0
5-7: visit
wolframalpha . com
or
rechneronline . de/function-graphs/
3 answers