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Chapter 3: Problem 44

a. Use the Leading Coefficient Test to determine the graph's end behavior. b. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. c. Find the \(y\) -intercept. d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=x^{4}-x^{2}$$

Short Answer

Expert verified
The graph of the function \(f(x) = x^{4} - x^{2}\) rises to the left and right, intersects the \(x\)-axis at \(-1, 0, 1\), intersects the \(y\)-axis at \(0\), and is symmetric about the \(y\)-axis. The function graph resembles the letter 'W'.

Step by step solution

01

Determining the end behavior

The degree of the polynomial \(f(x) = x^{4} - x^{2}\) is 4, which is an even number and the leading coefficient (the coefficient of \(x^{4}\)) is positive. According to the leading coefficient test, when the degree is positive and the leading coefficient is positive, the graph rises to the left and rises to the right.
02

Find the x-intercepts

To determine the x-intercepts, set the function equal to zero and solve for \(x\), giving us \(x^{4} - x^{2} = 0 \)\nSolving this we get \(x = 0, x = 1, x = -1\). As the coefficient of the highest degree term is positive, the graph cuts the x-axis at these points.
03

Find the y-intercept

Setting \(x = 0\) to find the \(y\)-intercept gives us \(f(0) = (0)^{4} - (0)^{2} = 0\). Therefore, the function intersects the \(y\)-axis at 0.
04

Check for symmetry

Plugging \(-x\) into the function we find \(f(-x) = (-x)^{4} - (-x)^{2} = x^{4} - x^{2} = f(x)\). This means that the graph has \(y\)-axis symmetry. The function does not have origin symmetry because \(f(-x)\) is not equal to \(-f(x)\).
05

Graphing the function

To plot the function, we know that it has \(x\)-intercepts at \(-1, 0, 1\) and \(y\)-intercept at 0. Moreover, it is symmetric about the \(y\)-axis. The function can be drawn resembles the letter 'W'. We can verify the accuracy of our graph by making sure it has no more than \(n - 1 = 4 - 1 = 3\) turning points, where \(n\) is the degree of the polynomial.

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Most popular questions from this chapter

In this exercise, we lead you through the steps involved in the proof of the Rational Zero Theorem. Consider the polynomial equation $$a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0}=0$$ and let \(\frac{P}{q}\) be a rational root reduced to lowest terms. a. Substitute \(\frac{p}{q}\) for \(x\) in the equation and show that the equation can be written as $$a_{n} p^{n}+a_{n-1} p^{n-1} q+a_{n-2} p^{n-2} q^{2}+\cdots+a_{1} p q^{n-1}=-a_{0} q^{n}$$ b. Why is \(p\) a factor of the left side of the equation? c. Because \(p\) divides the left side, it must also divide the right side. However, because \(\frac{P}{q}\) is reduced to lowest terms, \(p\) and \(q\) have no common factors other than \(-1\) and 1 Because \(p\) does divide the right side and has no factors in common with \(q^{n},\) what can you conclude? d. Rewrite the equation from part (a) with all terms containing \(q\) on the left and the term that does not have a factor of \(q\) on the right. Use an argument that parallels parts (b) and (c) to conclude that \(q\) is a factor of \(a_{n}\).

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