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Chapter 2: Problem 86

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the real zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. $$h(x)=\frac{1}{3} x^{3}(x-4)^{2}$$

Short Answer

Expert verified
The graph of the function \(h(x) = \frac{1}{3} x^{3}(x-4)^{2}\) will have x-intercepts at (0,0) and (4,0), fall to the left of the \(y-axis\), rise to the right, and pass through the calculated solution points.

Step by step solution

01

Apply the Leading Coefficient Test

The leading coefficient refers to the coefficient of the term with the highest power in the polynomial. Here, the leading coefficient is 1/3 and the degree (highest power) is 5. Since the coefficient is positive and the degree is odd, the graph of the polynomial falls to the left and rises to the right.
02

Find the Real Zeros

Also known as roots, the real zeros are the solutions to the equation \(h(x) = 0\), which occur where the graph of the function crosses the x-axis. From the equation \(h(x) = \frac{1}{3} x^{3}(x-4)^{2} = 0\), we get \(x = 0\) and \(x = 4\) as the real zeros. This means the curve crosses the x-axis at points (0,0) and (4,0).
03

Solve for Additional Points

Choose several x-values around each of the x-intercepts and calculate the corresponding y-values. For instance, choosing \(x = -1, 1, 2, 3, 5\) would reveal some points on the left and right of the zeros. For each x, calculate \(h(x) = \frac{1}{3} x^{3}(x-4)^{2}\) to get the respective y-values.
04

Draw the Graph

Plot the zeros and additional points on a graph and draw a smooth, continuous curve through all the points. Remember that the graph should fall to the left of the \(y-axis\) and rise to the right.

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