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Chapter 4: Problem 61

In each part: (i) Make a conjecture about the behavior of the graph in the vicinity of its \(x\) -intercepts. (ii) Make a rough sketch of the graph based on your conjecture and the limits of the polynomial as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\). (iii) Compare your sketch to the graph generated with a graphing utility. $$ \begin{array}{l}{\text { (a) } y=x(x-1)(x+1) \quad \text { (b) } y=x^{2}(x-1)^{2}(x+1)^{2}} \\ {\text { (c) } y=x^{2}(x-1)^{2}(x+1)^{3} \text { (d) } y=x(x-1)^{5}(x+1)^{4}}\end{array} $$

Short Answer

Expert verified
Sketch shows crossing/touching behavior at x-intercepts and correct end behaviors. Compare with graph utility for accuracy.

Step by step solution

01

Identifying x-intercepts for each function

For each polynomial, set it equal to zero to find the x-intercepts. \(y = x(x-1)(x+1)\) has x-intercepts at \(x=0, x=1, x=-1\). For \(y = x^2(x-1)^2(x+1)^2\), \(x = 0, x = 1, x = -1\), each repeated with multiplicity 2. For \(y = x^2(x-1)^2(x+1)^3\), \(x = 0, x = 1, x = -1\), where \(x = -1\) has multiplicity 3. Lastly, for \(y = x(x-1)^5(x+1)^4\), \(x = 0, x = 1, x = -1\), with \(x = 1\) having multiplicity 5 and \(x = -1\) having multiplicity 4.
02

Making conjectures about graph behavior near x-intercepts

For (a), each intercept is simple and the graph crosses the x-axis at each one. For (b), all intercepts have even multiplicity, so the graph touches but does not cross the x-axis. For (c), the graph touches at \(x=0\) and \(x=1\), and crosses at \(x=-1\). For (d), the graph crosses at \(x=0\), touches at \(x=1\) but does not cross, and at \(x=-1\) the graph makes a flat-topped tangent approach.
03

Analyzing end behavior of each polynomial

Each polynomial's leading term dictates the end behavior. For (a), it behaves like \(x^3\), rising to positive infinity as \(x \to \infty\) and falling to negative infinity as \(x \to -\infty\). For (b), it acts like \(x^6\), rising to positive infinity in both directions. (c) behaves like \(x^7\), resulting in the same end behavior as (a). Finally, (d) behaves like \(x^{10}\), similar to (b), rising in both directions.
04

Drawing rough sketches

Using the intercepts and end behaviors, sketch each graph: (a) starts negative, crosses each intercept. (b) starts positive, touches intercepts. (c) like (a), but dips sharply at \(x=-1\). (d) starts positive, touches \(x=1\), rises again.
05

Comparing to graphing utility results

Use a graphing calculator or software to plot each polynomial and check the shapes and behaviors at the intercepts and end behaviors. Compare these plots against your sketches to note accuracy or discrepancy.

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Key Concepts

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

x-intercepts
The x-intercepts of a polynomial function are the points where the graph crosses or touches the x-axis. These are found by setting the polynomial function equal to zero and solving for the variable x. In the context of the given polynomials:
  • For the first polynomial, \(y = x(x-1)(x+1)\), the x-intercepts are at \(x = 0, x = 1, x = -1\). Each of these intercepts occurs with multiplicity 1, meaning the graph will cross the x-axis at each of these points.
  • In the second polynomial, \(y = x^2(x-1)^2(x+1)^2\), there are also intercepts at \(x = 0, x = 1, x = -1\). However, because each intercept has even multiplicity, the graph will touch the x-axis and bounce back without crossing it at these points.
  • For \(y = x^2(x-1)^2(x+1)^3\), the intercepts remain the same but with a twist: \(x = -1\) has a multiplicity of 3, indicating the graph will cross at this point, while the others will just touch.
  • Finally, in \(y = x(x-1)^5(x+1)^4\), the graph crosses at \(x = 0\), gently touches at \(x = 1\) due to the higher multiplicity of 5, and makes a flatter touch at \(x = -1\).
Understanding x-intercepts helps to predict how the graph interacts with the x-axis.
graph behavior
Graph behavior near x-intercepts significantly affects the shape of the polynomial's graph. Here's a quick guide:
  • At points where the polynomial has intercepts with odd multiplicity, such as at \(x = 0\) for \(y = x(x-1)(x+1)\), the graph will cross the x-axis. This indicates a change in the signs of the y-values.
  • When intercepts have even multiplicity, the graph typically touches the x-axis but does not cross it. For example, in \(y = x^2(x-1)^2(x+1)^2\), all intercepts are touched due to their multiplicity being 2.
  • A larger odd multiplicity, like the 3 at \(x = -1\) in \(y = x^2(x-1)^2(x+1)^3\), results in a sharper crossing or a steeper section of the graph.
  • Significant multiplicity, such as 5, impacts the curve curvature, providing a tangent-like, flatter look on touching, as seen in \(x = 1\) for \(y = x(x-1)^5(x+1)^4\).
Recognizing these behaviors aids in accurately predicting and sketching graph features.
end behavior
End behavior describes how the graph behaves as \(x\) approaches positive or negative infinity. This behavior is determined by the polynomial's leading term, which significantly influences the overall graph's direction:
  • For \(y = x(x-1)(x+1)\), the end behavior resembles \(x^3\), meaning the graph dictates a downward slope for negative infinity and an upward climb toward positive infinity.
  • In \(y = x^2(x-1)^2(x+1)^2\), the function behaves like \(x^6\). This results in the graph rising to positive infinity in both directions, as even powers lead the function.
  • With \(y = x^2(x-1)^2(x+1)^3\), the leading term is \(x^7\). Like the cubic term in \(y = x(x-1)(x+1)\), this leads to falling and rising ends, reflecting the odd degree's influence.
  • For \(y = x(x-1)^5(x+1)^4\), the resemblance to \(x^{10}\) indicates upward trends from both sides, typical of even power behavior.
Understanding end behavior provides clarity on graph trends far away from the origin.
sketching
Sketching polynomial graphs can be a fun and insightful exercise once you grasp the essentials of x-intercepts, multiplicity, graph behavior, and end behavior. Here's how to proceed:
  • Firstly, plot the x-intercepts where the polynomial equals zero. Mark these with distinct points, knowing where the graph will touch or cross the x-axis.
  • Evaluate the multiplicity at each intercept to determine if the graph will have a sharp crossing or a smooth bounce.
  • Consider the end behavior to define how the graph will move towards infinity in both directions.
  • Finally, connect the intercepts smoothly, adjusting the graph's slope and curves based on multiplicity and leading terms.
By following these steps, you can produce rough yet highly informative sketches of polynomial behavior.
graphing utility
A graphing utility is an essential tool for visual learners and complex polynomial functions. It enables precise plotting and visual verification of a polynomial's graph characteristics. Here鈥檚 how they assist:
  • They automate finding x-intercepts and provide confirmations by displaying the exact points where the graph meets or approaches the x-axis.
  • Graphing utilities offer a thorough analysis of behavior near the x-intercepts by visually confirming whether curves cross or merely touch.
  • End behavior is vividly displayed, allowing for better comprehension of how graphs trend toward infinity.
  • Easily adjust visual accuracy; you can zoom in on particular sections of the graph to study finer details.
Using a graphing utility is like having an interactive sketchbook for polynomials, perfect for comparing hypothesized sketches with actual results.

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Most popular questions from this chapter

Let \(f(x)=x^{3}-4 x\) (a) Find the equation of the secant line through the points \((-2, f(-2))\) and \((1, f(1))\). (b) Show that there is only one point \(c\) in the interval \((-2,1)\) that satisfies the conclusion of the Mean-Value Theorem for the secant line in part (a). (c) Find the equation of the tangent line to the graph of \(f\) at the point \((c, f(c)) .\) (d) Use a graphing utility to generate the secant line in part (a) and the tangent line in part (c) in the same coordinate system, and confirm visually that the two lines seem parallel.

Verify that the hypotheses of Rolle's Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$ f(x)=\cos x ;[\pi / 2,3 \pi / 2] $$

If \(f\) is a periodic function, then the locations of all absolute extrema on the interval \((-\infty,+\infty)\) can be obtained by finding the locations of the absolute extrema for one period and using the periodicity to locate the rest. Use this idea in these exercises to find the absolute maximum and minimum values of the function, and state the \(x\) -values at which they occur. \(f(x)=3 \cos \frac{x}{3}+2 \cos \frac{x}{2}\)

Verify that the hypotheses of the Mean-Value Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$ f(x)=x^{2}-x ;[-3,5] $$

Newton's Method uses a local linear approximation to \(y=f(x)\) at \(x=x_{n}\) to find an "improved" approximation \(x_{n+1}\) to a zero of \(f\). Your friend proposes a process that uses a local quadratic approximation to \(y=f(x)\) at \(x=x_{n}\) (that is, matching values for the function and its first two derivatives) to obtain \(x_{n+1} .\) Discuss the pros and cons of this proposal. Support your statements with some examples.
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