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Chapter 5: Problem 47

For the following exercises, graph the polynomial functions. Note \(x\) -and \(y\) -intercepts, multiplicity, and end behavior. $$ n(x)=-3 x(x+2)(x-4) $$

Short Answer

Expert verified
The graph of \(-3x(x+2)(x-4)\) crosses at x-intercepts -2, 0, 4, it starts from positive when x is negative, goes to infinity negatively, and ends negatively when x is positive.

Step by step solution

01

Understanding the Function

The given polynomial is \( n(x) = -3x(x+2)(x-4) \). This indicates it is a cubic polynomial. Before moving to the graph, let's analyze its components: roots, multiplicities, and end behavior.
02

Finding the Roots and Multiplicities

The roots can be found by setting each factor equal to zero: \( x = 0 \), \( x+2 = 0 \Rightarrow x = -2 \), and \( x-4 = 0 \Rightarrow x = 4 \). Each root has a multiplicity of 1 because each factor appears once.
03

Determining the Y-intercept

To find where the graph intersects the y-axis, evaluate \( n(x) \) at \( x = 0 \): \( n(0) = -3(0)(0+2)(0-4) = 0 \). Thus, the y-intercept is \( (0, 0) \).
04

Analyzing End Behavior

The end behavior is determined by the leading term when the polynomial is expanded, which is \( -3x^3 \). As \( x \to \infty \), \( n(x) \to -\infty \); and as \( x \to -\infty \), \( n(x) \to \infty \). This means the right end of the graph goes downward, and the left end goes upward.
05

Plotting the X-intercepts and Y-intercept

Plot the x-intercepts at \( x = -2, 0, 4 \) and the y-intercept at \( (0, 0) \) on a coordinate plane. All x-intercepts are also points on the x-axis for the function graph.
06

Sketching the Graph

Using the intercepts and end behavior, sketch the graph by starting slightly above the x-axis at \( x = -fty \), coming through the x-axis at \( x = -2 \), rising and passing through \( (0,0) \), and finally falling and passing through \( x = 4 \) before continuing downwards.

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Key Concepts

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

Polynomial Roots
Polynomial roots, also known as zeros, are values of x where the polynomial equation equals zero. For the function \( n(x) = -3x(x+2)(x-4) \), the roots can be identified by setting each factor of the polynomial to zero:
  • \( x = 0 \)
  • \( x+2 = 0 \; \Rightarrow \; x = -2 \)
  • \( x-4 = 0 \; \Rightarrow \; x = 4 \)
These solutions, \( x = 0, -2, \) and \( 4 \), are the points where the graph of the polynomial crosses the x-axis.
Finding the roots is a critical step in sketching the graph of any polynomial function, as these intercepts provide foundational points at which the function changes direction when crossing the x-axis.
End Behavior
The end behavior of a polynomial function describes how the graph behaves as \( x \) approaches positive or negative infinity. For the polynomial \( n(x) = -3x(x+2)(x-4) \), which is a cubic function, the leading term is \( -3x^3 \).
A look at the leading term helps determine the overall direction of the polynomial ends:
  • As \( x \to \infty \), because of the negative coefficient in the leading term \( -3x^3 \), \( n(x) \to -\infty \)
  • As \( x \to -\infty \), \( n(x) \to \infty \)
This implies that the graph dips downwards as x becomes very large in the positive direction and upwards when it goes in the negative direction.
Understanding a function's end behavior helps in predicting and sketching the graph beyond the immediate vicinity of x- and y-intercepts.
Multiplicity
Multiplicity refers to how many times a particular root appears in a polynomial function. In our polynomial \( n(x) = -3x(x+2)(x-4) \), each factor \( x, \; (x+2), \; (x-4) \) appears only once, meaning each root has a multiplicity of 1.
When a root has a multiplicity of:
  • 1 - The graph will cross the x-axis at this point
  • 2 - The graph will touch the x-axis and turn around
  • 3 or more - The graph will pass through but tends to flatten at the x-axis
In our case, since all roots have a multiplicity of 1, the graph will cross the x-axis at each x-intercept distinctly at \( x = -2, 0, \text{and} \; 4 \). Knowing multiplicity allows you to predict how the graph behaves around the roots.
Y-intercept
The y-intercept of a polynomial function is where the graph crosses the y-axis. It can be found by evaluating the function at \( x = 0 \). For our polynomial \( n(x) = -3x(x+2)(x-4) \):\[n(0) = -3(0)(0+2)(0-4) = 0\]Thus, the y-intercept is \((0, 0)\).
The y-intercept acts as both an x- and a y-intercept when it is located at the origin for a polynomial function. This point provides crucial information for sketching the graph and serves as a landmark showing where the curve crosses the y-axis. Understanding and identifying the y-intercept simplifies the plotting process, giving you a starting point or a connector between multiple intercepts.

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Most popular questions from this chapter

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