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Chapter 3: Problem 22

Analyzing the Graph of a Function Exercises \(9-36,\) analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$f(x)=\frac{x^{3}}{x^{2}-9}$$

Short Answer

Expert verified
The graph of \(f(x) = \frac{x^3}{x^2-9}\) has an x-intercept and y-intercept at (0,0), vertical asymptotes at x = -3 and x = 3, a local minimum at x=0, and points of inflection at x = -2 and x = 2.

Step by step solution

01

Find x-intercepts and y-intercepts

Set \(f(x) = 0\) to find x-intercepts. The equation becomes \(x^3 = 0\) which gives x = 0. For y-intercept, put \(x = 0\) in \(f(x)\), this gives y = 0. So, the intercepts point is (0,0)
02

Find Vertical and Horizontal Asymptotes

Factor the denominator \(x^2 - 9\) to \((x-3)(x+3)\). Setting it to zero gives the vertical asymptotes x = -3 and x = 3. For horizontal asymptote, since degree of numerator > degree of denominator, there is no horizontal asymptote.
03

Find Extrema

Find the first derivative of \(f(x)\) to identify any local maxima or minima. The first derivative of \(f(x) = \frac{x^3}{x^2-9}\) simplifies after differentiation as \(f'(x) = \frac{9x}{(x^2-9)^2}\). Setting \(f'(x) = 0\), we find x = 0 to be an extremum point.
04

Find Points of Inflection

Find the second derivative of \(f(x)\), and set it to zero. The second derivative \(f''(x) = \frac{18(x^2-4)}{(x^2-9)^3}\). Solving \(f''(x) = 0\), gives x = -2 and 2. Thus, we have points of inflection at x = -2 and x = 2.
05

Analyse and Sketch the Graph

Join the identified points and vertical asymptotes to sketch the graph. Use a graphing utility to verify the results.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Intercepts
Intercepts are key points where a graph crosses the axes. These give valuable insight into the behavior of the function at specific points.
  • X-intercepts: These occur when the function value becomes zero (\(f(x) = 0\)). For the given function \(f(x) = \frac{x^3}{x^2-9}\), setting \(x^3 = 0\) finds the x-intercept at \(x = 0\).
  • Y-intercepts: These are found by evaluating the function when \(x = 0\). inserting this into \(f(x)\) also gives the y-intercept at \(y = 0\).

Thus, the coordinate (0, 0) is where both intercepts occur, offering an intuitive starting point for understanding the graph's placement in the coordinate plane.
Asymptotes
Asymptotes guide us in recognizing how the graph behaves towards the extremes or undefined points. They can be vertical, horizontal, or oblique.
  • Vertical Asymptotes: These occur where the function tends to infinity. They appear when the denominator of a fraction tends towards zero while the numerator does not. By factoring the denominator \(x^2 - 9\) to \((x-3)(x+3)\), we find vertical asymptotes at \(x = -3\) and \(x = 3\)
  • Horizontal Asymptotes: These reflect the function's behavior as \(x\) approaches infinity. However, if the function's numerator degree exceeds the denominator, like our function, there are no horizontal asymptotes.

Understanding asymptotes is crucial in depicting the graph in extreme limits and where the function could not possibly reach a definite value.
Extrema
Extrema reveal the highest or lowest points in a function's region. They assist us in understanding the peaks and troughs of a graph.
  • Local Maximum and Minimum: Found by calculating the first derivative of the function \(f'(x) = \frac{9x}{(x^2-9)^2}\). Setting this equal to 0 gives us the extremum point at \(x = 0\).

At this point, the graph experiences either a peak (maximum) or a trough (minimum). Additionally, extrema offer a critical analysis of the function鈥檚 varying nature throughout its domain, guiding us to more meaningful graph interpretations.
Points of Inflection
Points of inflection are where the curve shifts its concavity, providing insights into the nature and direction of the curve changes.
  • Detecting Inflection Points: By finding the second derivative \(f''(x) = \frac{18(x^2-4)}{(x^2-9)^3}\) and setting it to 0, we identify points where the function's concavity changes at \(x = -2\) and \(x = 2\).

These points mark significant shifts in the graph's shape, allowing students to predict how sections of the graph will curve or straighten. Recognizing these points provides a full picture of the graph's dynamics, enhancing both your understanding and the graphical representation.

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