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Chapter 4: Problem 22

\(15-26\) Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=\frac{1}{4}(x+1)^{3}(x-3) $$

Short Answer

Expert verified
Degree 4 polynomial with roots at \(-1\) and \(3\), y-intercept at \((0, -\frac{3}{4})\); end behavior: rises to infinity at both ends.

Step by step solution

01

Analyze the Degree and Leading Coefficient

The polynomial given is \( P(x) = \frac{1}{4}(x+1)^3(x-3) \). To determine the degree of the polynomial, add up the exponents of the factors: \((x+1)^3\) contributes 3 and \(x-3\) contributes 1, giving us a total degree of 4. The leading coefficient is \(\frac{1}{4}\), which is positive. This tells us the overall shape will have similar end behavior to a degree 4 polynomial, opening upwards on both sides.
02

Determine the Roots

To find the x-intercepts (roots), set \( P(x) = 0 \). The roots are derived from the factors: \((x+1)\) gives \(x = -1\) with multiplicity 3, and \((x-3)\) gives \(x = 3\) with multiplicity 1. Thus, the roots of the polynomial are \(x = -1\) and \(x = 3\).
03

Identify the Behavior at Intercepts

At \(x = -1\), the root has an odd multiplicity of 3, which implies that the graph will cross the x-axis at this point but will flatten out as it does so. At \(x = 3\), having multiplicity 1, the graph will also cross the x-axis, but with a linear character, meaning it will not flatten at this intercept.
04

Calculate the Y-Intercept

To find the y-intercept, evaluate \( P(x) \) at \( x = 0 \). Plugging this in: \( P(0) = \frac{1}{4}(0+1)^3(0-3) = \frac{1}{4}(1)(-3) = -\frac{3}{4} \). Thus, the y-intercept is \((0, -\frac{3}{4})\).
05

Sketch the Graph

Use the information from the previous steps to sketch the graph. Start by plotting the roots \(x = -1\) and \(x = 3\), and the y-intercept \((0, -\frac{3}{4})\). Considering the degree and leading coefficient, the end behavior of this 4th-degree polynomial means the graph will rise to positive infinity as \(x\) goes to both \(+\infty\) and \(-\infty\). The graph will cross the x-axis at both intercepts and behave as described in Step 3 around these critical points.

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Key Concepts

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

Polynomial End Behavior
The end behavior of a polynomial function provides insight into how the graph behaves as the variable \(x\) approaches infinity (\(+\infty\)) and negative infinity (\(-\infty\)). The most significant influences on end behavior are the degree of the polynomial and the leading coefficient. For the polynomial \( P(x) = \frac{1}{4}(x+1)^3(x-3) \), with a degree of 4 and a positive leading coefficient of \(\frac{1}{4}\):
  • Since the degree (4) is even, both ends of the graph will point in the same direction.
  • A positive leading coefficient suggests that the graph will rise as \(x\) moves towards both \(+\infty\) and \(-\infty\).
These factors together lead to an upward "U" shape for the polynomial's end behavior.
Roots of Polynomials
Roots of a polynomial, often referred to as zeros or x-intercepts, are values of \(x\) at which the polynomial evaluates to zero. To find the roots, set the polynomial equation equal to zero and solve for \(x\).For the polynomial \( P(x) = \frac{1}{4}(x+1)^3(x-3) \):
  • From the factor \((x+1)^3\), the root is \(x = -1\).
  • From the factor \((x-3)\), the root is \(x = 3\).
These roots indicate the points where the graph of the polynomial will touch or cross the x-axis. Identifying these points helps in sketching the polynomial accurately.
Multiplicity of Roots
Multiplicity refers to the number of times a particular root appears. It affects how the graph behaves at the corresponding intercept. The structure of the polynomial shows these multiplicities explicitly.In \( P(x) = \frac{1}{4}(x+1)^3(x-3) \):
  • The root \(x = -1\) has a multiplicity of 3, which means the graph will cross but flatten at the x-axis, showing a cubic-like behavior around \(-1\).
  • The root \(x = 3\) has a multiplicity of 1, indicating that the graph crosses the x-axis in a linear fashion at this point.The multiplicities inform whether the graph touches or crosses the x-axis and the nature of this interaction, which is crucial for graph sketching.
Intercepts in Algebra
Intercepts provide essential anchor points for sketching graphs. The primary intercepts in algebra include x-intercepts (roots) and the y-intercept.For the polynomial \( P(x) = \frac{1}{4}(x+1)^3(x-3) \):
  • X-intercepts: Root analysis gives x-intercepts at \((x, y) = (-1, 0)\) and \((3, 0)\). These indicate where the graph will set at zero on the x-axis.
  • Y-intercept: Found by setting \(x = 0\) which gives \(P(0) = -\frac{3}{4}\), so the y-intercept is \((0, -\frac{3}{4})\).
These intercepts assist not only in placing the graph on coordinate axes but in understanding the overall graph shift and shape.

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Most popular questions from this chapter

A Rational Function with No Asymptote Explain how you can tell (without graphing it) that the function $$ r(x)=\frac{x^{6}+10}{x^{4}+8 x^{2}+15} $$ has no \(x\) -intercept and no horizontal, vertical, or slant asymptote. What is its end behavior?'

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=3 x^{3}+17 x^{2}+21 x-9 $$

Graph the rational function, and find all vertical asymptotes, x- and y-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$ y=\frac{x^{4}}{x^{2}-2} $$

Graph the rational function \(f,\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$ f(x)=\frac{x^{3}-2 x^{2}+16}{x-2}, g(x)=x^{2} $$

Use Descartes鈥� Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$ P(x)=x^{5}+4 x^{3}-x^{2}+6 x $$
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