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Chapter 3: Problem 70

Sketch the function. \(m(x)=0.1(x-3)^{2}(x+1)^{3}\)

Short Answer

Expert verified
Roots at \( x = 3 \) and \( x = -1 \) with specified multiplicities. End behavior: \( \infty \) as \( x \rightarrow \infty \) and \( -\infty \) as \( x \rightarrow -\infty \).

Step by step solution

01

Identify the roots

The roots are the values of x for which the function equals zero. Set the function to zero and solve for x: \[ 0 = 0.1(x - 3)^2(x + 1)^3 \] The roots are: \( x = 3 \) and \( x = -1 \).
02

Determine the multiplicity of each root

The multiplicity of a root is the exponent of the factor. The root \( x = 3 \) comes from \( (x - 3)^2 \) and has a multiplicity of 2. The root \( x = -1 \) comes from \( (x + 1)^3 \) and has a multiplicity of 3.
03

Determine the end behavior

Since the leading term is \(0.1(x-3)^2(x+1)^3\), which simplifies to \(0.1x^5\), the highest power of x is 5. The leading coefficient is positive (0.1), so as \( x \rightarrow \infty \), \( m(x) \rightarrow \infty \), and as \( x \rightarrow -\infty \), \( m(x) \rightarrow -\infty \).
04

Analyze the behavior at each root

For \( x = 3 \), with a multiplicity of 2, the graph touches the x-axis and turns around. For \( x = -1 \), with a multiplicity of 3, the graph crosses the x-axis and changes direction.
05

Find additional points if needed

Calculate a few additional points to get a better idea of the graph shape. For example: \( m(0) = 0.1(0-3)^2(0+1)^3 = 0.1(9)(1) = 0.9 \) \( m(1) = 0.1(1-3)^2(1+1)^3 = 0.1(4)(8) = 3.2 \)
06

Sketch the graph

Combine all this information: the roots with their behaviors, the end behavior, and any additional points to sketch the function. The graph starts from \( -\u221e \) going to \( -\u221e \), touches at \( x=3 \), and crosses at \( x=-1 \).

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Key Concepts

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

Roots of Polynomial
The roots of a polynomial are the values of x that make the polynomial equal to zero. In simpler terms, they're the points where the graph of the polynomial touches or crosses the x-axis. To find the roots, we set the polynomial function equal to zero and solve for x. For example, in the polynomial function \(m(x)=0.1(x-3)^{2}(x+1)^{3}\), setting \(0.1(x-3)^{2}(x+1)^{3} = 0\) allows us to find the roots. The roots are \(x = 3\) and \(x = -1\). These are the points where the function graph will interact with the x-axis.
Multiplicity of Roots
The multiplicity of a root refers to how many times that root is repeated in the polynomial. It is determined by the exponent of the factor corresponding to that root. For the function \(m(x)=0.1(x-3)^{2}(x+1)^{3}\), the root \(x = 3\) comes from the term \((x-3)^{2}\) and has a multiplicity of 2. This means the graph touches the x-axis at this point and turns around. Similarly, the root \(x = -1\) comes from the term \((x+1)^{3}\) and has a multiplicity of 3. This means the graph crosses the x-axis and changes direction at this point. Understanding multiplicity helps us predict the behavior of the graph at each root.
End Behavior of Polynomial Functions
The end behavior of a polynomial function is how the graph behaves as x approaches infinity \((\rightarrow \rightarrow\) or negative infinity \((-\rightarrow \rightarrow\). It is largely determined by the polynomial's leading term. For the function \(m(x)=0.1(x-3)^{2}(x+1)^{3}\), the leading term is found by multiplying out the highest powers of x from each factor: \(0.1x^{5}\).
Since the exponent (5) is odd and the leading coefficient (0.1) is positive, the end behavior is:
  • As x \rightarrowinfty, m(x) \rightarrowinfty
  • As x -\rightarrowinfty, m(x) -\rightarrowinfty
This means that as x gets very large (positively or negatively), the function value (y) will trend towards \( \rightarrow \rightarrow\) and \(-\rightarrow \rightarrow\), respectively.
Graphing Polynomial Functions
Graphing polynomial functions involves combining information about the roots, their multiplicity, and the end behavior. For \(m(x)=0.1(x-3)^{2}(x+1)^{3}\), we:
  • Identify the roots: \(x = 3\) and \(x = -1\).
  • Note the multiplicity: Root \(x = 3\) has a multiplicity of 2 (graph touches and turns around), and root \(x = -1\) has a multiplicity of 3 (graph crosses x-axis and changes direction).
  • Consider end behavior: As x \rightarrowinfty, m(x) \rightarrowinfty and as x -\rightarrowinfty, m(x) -\rightarrowinfty.
  • Calculate additional points: For example, \( m(0) = 0.9\) and \(m(1) = 3.2\) to help shape the graph.
Combining this information helps to sketch a complete and accurate graph of the polynomial function.

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Most popular questions from this chapter

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