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Chapter 5: Problem 42

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

Short Answer

Expert verified
The polynomial intersects x-axis at \(-3\) and \(2\) with respective multiplicities 2 and 1, y-intercept at \(-18\), and ends with behavior: left to \(-\infty\), right to \(\infty\).

Step by step solution

01

Identify the polynomial's degree

The polynomial is given by \( f(x) = (x+3)^2(x-2) \). Expand it to identify the degree: \((x+3)^2(x-2) = (x^2 + 6x + 9)(x-2)\). Further expansion gives \(x^3 + 4x^2 - 3x - 18\). The degree of the polynomial is 3, indicating that the end behavior will resemble that of \(x^3\).
02

Find the x-intercepts and their multiplicities

Set \( f(x) = (x+3)^2(x-2) = 0 \) to find the x-intercepts. The equation implies \((x+3)^2 = 0\) or \((x-2) = 0 \). This gives x-intercepts at \(x=-3\) and \(x=2\). The intercept at \(x=-3\) has a multiplicity of 2 (since it's squared), and the intercept at \(x=2\) has a multiplicity of 1.
03

Find the y-intercept

To find the y-intercept, evaluate the function at \(x=0\). Substitute \(x=0\) into the function: \(f(0) = (0+3)^2(0-2) = 9(-2) = -18\). So, the y-intercept is \((0, -18)\).
04

Determine the effect of multiplicity on the intercepts

The x-intercept at \(x = -3\) with multiplicity 2 means the graph will touch the x-axis and bounce back. The intercept at \(x = 2\) with multiplicity 1 means the graph will pass through the x-axis.
05

Analyze the end behavior

Since the polynomial is of degree 3 (odd), the ends of the graph will have opposite behavior. The leading term after expansion is \(x^3\), indicating that as \(x \to \infty\), \(f(x) \to \infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\).
06

Graph the polynomial using the information

Plot the x-intercepts at \(x=-3\) and \(x=2\), and the y-intercept at \(y=-18\). Use the multiplicity information: the graph touches the x-axis and bounces at \(x=-3\), and crosses the x-axis at \(x=2\). Sketch the end behavior where the left end goes to \(-\infty\) and the right end goes to \(\infty\).

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Key Concepts

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

X-intercepts
In polynomial graphing, finding the x-intercepts involves determining where the graph crosses or touches the x-axis. An x-intercept occurs when the value of the function is zero, which translates to solving the equation where the polynomial equals zero.
  • The given polynomial is \( f(x) = (x+3)^2(x-2) \).
  • To find the x-intercepts, set the equation equal to zero: \( (x+3)^2(x-2) = 0 \).
  • This gives us solutions at \( x = -3 \) and \( x = 2 \) as these values make the function zero.
Each intercept can occur at a point where the graph either crosses or just touches the axis, depending on other factors such as its multiplicity.
Y-intercepts
The y-intercept of a polynomial is the point where the graph of the function crosses the y-axis. To find the y-intercept, we evaluate the function at \( x = 0 \).
  • For \( f(x) = (x+3)^2(x-2) \), substitute \( x = 0 \).
  • Calculate \( f(0) = (0+3)^2(0-2) = 9(-2) = -18 \).
  • Therefore, the y-intercept is at the point \( (0, -18) \).
Identifying this point is important for graphing as it provides a key location on the y-axis.
Multiplicity
Multiplicity illustrates how a polynomial touches or crosses the x-axis at any of its intercepts. It refers to the number of times a particular root appears in the polynomial.
  • In \( f(x) = (x+3)^2(x-2) \), the root \( x = -3 \) appears twice because of the squared term, giving it a multiplicity of 2.
  • Meanwhile, \( x = 2 \) is a simple root, appearing only once, with a multiplicity of 1.
  • An intercept with an odd multiplicity indicates the graph will cross the x-axis at that point (as we see at \( x = 2 \)).
  • Even multiplicities, like at \( x = -3 \), demonstrate the graph touches the x-axis and "bounces" off it without crossing through.
Understanding multiplicity helps in predicting the behavior of the graph at specific intercept points, making polynomial graphing easier to anticipate.
End Behavior
End behavior describes what happens to the graph of a polynomial as we move towards the extreme ends of the x-axis. It's primarily influenced by the degree and leading coefficient of the polynomial.
  • The polynomial \( f(x) = x^3 + 4x^2 - 3x - 18 \) is of degree 3, which is odd.
  • This means that the graph will behave oppositely on either side鈥� as \( x \rightarrow \infty, f(x) \rightarrow \infty \) and as \( x \rightarrow -\infty, f(x) \rightarrow -\infty \).
  • The leading term here is \( x^3 \), setting the framework for the end behavior of the graph.
By understanding the degree and leading term, students can confidently anticipate the basic overall shape and direction of the polynomial graph's ends. Recognizing end behavior aids in correctly sketching the graph at the extremities.

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Most popular questions from this chapter

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with \(y\) -coordinates given. $$f(x)=x^{3}-x-2, y=1,2,3$$

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as \(x\) and \(z\) and inversely as \(w .\) When \(x=3\), \(z=5\), and \(w=6,\) then \(y=10\).

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as \(x\) and \(z\) and inversely as the square root of \(w\) and the square of \(t\). When \(x=3, z=1, w=25\), and \(t=2,\) then \(y=6\).

For the following exercises, determine the function described and then use it to answer the question. A container holds 100 \(\mathrm{ml}\) of a solution that is 25 \(\mathrm{ml}\) acid. If \(n \mathrm{ml}\) of a solution that is 60\(\%\) acid is added, the function \(C(n)=\frac{25+0.6 n}{100+n}\) gives the concentration, \(C,\) as a function of the number of ml added, \(n .\) Express \(n\) as a function of \(C\) and determine the number of \(m 1\) that need to be added to have a solution that is 50\(\%\) acid.

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies directly as the cube of \(x\) and when \(x=2, y=4\).
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