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.
I suggest look at the 7 steps in the text. I certainly have no idea what they are.
Evidently the accompanying essay describes the steps.
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/
R(x)=x^3-125/x^2-49
To analyze the graph of the function R(x) = x^2 - 49/x^4 - 16 using the 7 steps outlined in section 4.3, follow these steps:
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.
- The numerator x^2 can't be factored further, and the denominator x^4 - 16 can be factored as (x^2 - 4)(x^2 + 4).
- The domain of R is all real numbers except where the denominator equals zero. So we equate the denominator to zero: (x^2 - 4)(x^2 + 4) = 0. Solving this equation gives x = -2, x = 2.
- The y-intercept is obtained by evaluating R(0):
R(0) = (0^2 - 49)/(0^4 - 16) = -49/(-16) = 49/16 ≈ 3.0625
- Plot the y-intercept at the point (0, 3.0625).
Step 2: Write R in lowest terms as p(x)/q(x) and find the real zeros of the numerator. Determine the behavior of the graph of R near each x-intercept.
- R can be written as R(x) = (x^2 - 49)/(x^4 - 16).
- To find the real zeros of the numerator, solve the equation x^2 - 49 = 0. This gives x = -7, x = 7.
- Determine the behavior near each x-intercept:
- Near x = -7: The graph approaches from the left and goes up towards negative infinity.
- Near x = 7: The graph approaches from the right and goes up towards positive infinity.
- Plot the x-intercepts at (-7, 0) and (7, 0) and indicate the behavior near each intercept.
Step 3: Find the real zeros of the denominator. These determine the vertical asymptotes of the graph. Graph each vertical asymptote using a dashed line.
- The real zeros of the denominator are found by solving x^4 - 16 = 0. This gives x = -2 and x = 2.
- These values determine the vertical asymptotes at x = -2 and x = 2.
- Graph each vertical asymptote using dashed lines.
Step 4: Locate any horizontal or oblique asymptotes and determine if the graph intercepts them. Graph the asymptotes using dashed lines.
- For this step, we look at the degree of the numerator and denominator.
- The degree of the numerator (2) is less than the degree of the denominator (4).
- Therefore, the graph will have a horizontal asymptote at y = 0 (the x-axis).
- Plot this asymptote using a dashed line.
- To check for any point of intersection, solve the equation y = 0 and find the x-values.
- Set numerator equal to zero: x^2 - 49 = 0. This gives x = -7, x = 7.
- The graph intersects the x-axis at x = -7 and x = 7.
- Plot these points of intersection.
Step 5: Divide the x-axis into intervals using the real zeros of the numerator and denominator. Determine where the graph is above or below the x-axis.
- Divide the x-axis into four intervals: (-∞, -7), (-7, -2), (-2, 2), (2, ∞).
- Choose a test point in each interval and evaluate R there.
- Example:
- In the interval (-∞, -7), choose x = -8. Substitute it into R: R(-8) = (-8^2 - 49)/(-8^4 - 16) = (-64 - 49)/(-4096 - 16) = -113/ -4112 ≈ 0.0275.
- Since R(-8) is positive, the graph is above the x-axis in this interval.
- Repeat this process for the other intervals and determine where the graph is above or below the x-axis.
- Plot the points found.
Step 6: Analyze the behavior of the graph of R near each asymptote and indicate this behavior on the graph.
- The behavior of the graph near each vertical asymptote (x = -2 and x = 2) is important.
- As x approaches -2 or 2, the graph heads towards positive or negative infinity, depending on the intervals.
- Indicate this behavior on the graph near each asymptote.
Step 7: Put all the information together to obtain the graph of R.
- Plot all the points, including the x-intercepts, y-intercept, points of intersection of the graph and asymptotes, and the behavior near each point.
- Draw the graph following the behavior described above: approaching each x-intercept and asymptote.
- The graph should display the shape, direction, and key points based on the information gathered from the previous steps.
Follow these steps to analyze and graph the function R(x) = x^2 - 49/x^4 - 16.