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Chapter 3: Problem 86

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(g(x)=x^{4}-4 x^{2}\)

Short Answer

Expert verified
Using the above steps, we are able to determine that the polynomial function \(g(x) = x^{4} - 4x^{2}\) has zeros at \(x = 0, 2, -2\) and turns at those points. The graph approaches positive infinity as \(x\) approaches positive and negative infinity. By plotting these key points and drawing a curve through them, we can accurately depict the behavior of the function.

Step by step solution

01

Apply Leading Coefficient Test

The Leading Coefficient Test tells us about the end behavior of a polynomial function. In this function \(g(x) = x^{4} - 4x^{2}\), the leading term is \(x^{4}\). Because the degree of the polynomial (4) is even and the leading coefficient (1) is positive, the graph of the function rises as \(x\) approaches infinity and also rises as \(x\) approaches negative infinity.
02

Find the Zeros of the Function

The zeros or roots of a function are the x-values that make the function equal to zero. To find the zeros of the function \(g(x) = x^{4} - 4x^{2}\), set the function equal to zero and solve for \(x\):\n0 = \(x^{4} - 4x^{2}\)\nWe factor the quadratic to find the roots:\n0 = \(x^{2}(x^{2} - 4)\)\nThis gives us three roots: \(x = 0\), \(x=2\), and \(x = -2\).
03

Plot Solution Points

We know the function touches or crosses the x-axis at \(x = 0, 2, -2\). We can plot these three points on the graph. Furthermore, we can select more points for a more accurate graph. For example, when \(x=-3\), \(g(x) = 5\). Therefore, we can plot these four points (-3,5), (-2,0), (0,0) and (2,0).
04

Draw a continuous curve

Join the points with a smooth curve. Since the degree of polynomial is even and the leading coefficient is positive, the graph will go up towards positive infinity on both sides.

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Key Concepts

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

Leading Coefficient Test
The Leading Coefficient Test is a quick method to determine the end behavior of polynomial functions. It focuses on the leading term, which is the term with the highest exponent in the polynomial. Let's take a closer look at the function g(x)=x^{4}-4x^{2}. Here, the leading term is x^{4}, and the coefficient of this term is 1. Since the exponent (4) is even and the leading coefficient is positive, according to this test, the graph of the function will rise on both ends. This means as x tends to negative infinity and positive infinity, the function's values will become increasingly large. This affects how the graph is sketched, as we know it will start and end in the upper regions of the graph, giving it a 'U' shape.
Finding Zeros of a Polynomial
To find the zeros or roots of a polynomial, we want to determine the x values that make the polynomial equal to zero. These zeros are where the graph will intersect or touch the x-axis. For the function g(x)=x^{4}-4x^{2}, we set the function equal to zero and solve for x: 0 = x^{4} - 4x^{2}. Factoring out an x^{2} gives us 0 = x^{2}(x^{2} - 4). Factor further to get x^{2}, (x + 2), and (x - 2). This gives us three zeros: x = 0, x = -2, and x = 2. Understanding how to find these critical points is essential as they are the foundation for sketching the graph correctly.
Plotting Solution Points
Plotting solution points involves selecting values for x and calculating the corresponding g(x) to get points on the graph. It's important to choose a range of values, including values around the zeros identified earlier. For instance, we can pick x = -3 and find g(-3) = (-3)^4 - 4(-3)^2 = 81 - 36 = 45. Similarly, calculate g(x) for several other values of x to establish a pattern and aid in drawing the graph. These calculated points, along with the zeros, give us precise locations on the graph and help us visualize how the curve moves between these points.
Continuous Curve
Polynomial functions are represented by continuous curves without breaks, gaps, or sharp corners. When drawing the graph of a polynomial function such as g(x)=x^{4}-4x^{2}, you'll connect all plotted points with a smooth, flowing line. The graph should pass through the zeros at x = 0, x = -2, and x = 2. Since the function involves a fourth degree polynomial with positive leading coefficient, the tails of the graph will extend up towards positive infinity on both ends. Combining this understanding with the plotted points and zeros, we can sketch the complete graph, ensuring it reflects the continuous nature of polynomial functions.

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Most popular questions from this chapter

Find all real zeros of the polynomial function. $$f(x)=4 x^{4}-55 x^{2}-45 x+36$$

Match the cubic function with the correct number of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ;\) Irrational zeros: \(\mathbf{0}\) (c) Rational zeros: \(1 ; \quad\) Irrational zeros: 2 (d) Rational zeros: \(1 ;\) Irrational zeros: \(\mathbf{0}\) $$f(x)=x^{3}-x$$

Divide using long division. $$\left(4 x^{5}+3 x^{3}-10\right) \div(2 x+3)$$

Consider a physics laboratory experiment designed to determine an unknown mass. A flexible metal meter stick is clamped to a table with 50 centimeters overhanging the edge (see figure). Known masses \(M\) ranging from 200 grams to 2000 grams are attached to the end of the meter stick. For each mass, the meter stick is displaced vertically and then allowed to oscillate. The average time \(t\) (in seconds) of one oscillation for each mass is recorded in the table. $$\begin{array}{|c|c|} \hline \text { Mass, \(M\) } & \text { Time, \(t\) } \\ \hline 200 & 0.450 \\ 400 & 0.597 \\ 600 & 0.712 \\ 800 & 0.831 \\ 1000 & 0.906 \\ 1200 & 1.003 \\ 1400 & 1.088 \\ 1600 & 1.126 \\ 1800 & 1.218 \\ 2000 & 1.338 \\ \hline \end{array}$$ A model for the data is given by $$t=\frac{38 M+16,965}{10(M+5000)}$$ (a) Use the table feature of a graphing utility to create a table showing the estimated time based on the model for each of the masses shown in the table. What can you conclude? (b) Use the model to approximate the mass of an object when the average time for one oscillation is 1.056 seconds.

Determine algebraically any point(s) of intersection of the graphs of the equations. Verify your results using the intersect feature of a graphing utility. $$\begin{aligned} &y=9-x^{2}\\\ &y=x+3 \end{aligned}$$
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