Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

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. \(f(x)=3 x^{3}-24 x^{2}\)

Step-by-Step Solution

Answer

The final sketch of the function will have x-intercepts at x=0 and x=8. The plot will look like a smooth curve rise to the right and fall to the left, crossing the x-axis at the x-intercepts. All selected points will lie on this curve.

Step 1: Applying the Leading Coefficient Test

A polynomial function \(f(x) = a_n x^n + ... + a_1x +a_0\) terminates in \(a_n x^n\).The leading coefficient refers to \(a_n\) and \(n\) is the degree of the polynomial. If the leading coefficient is positive and the degree is odd, then the graph rises to the right and falls to the left. The function given here is \(3x^3 - 24x^2\). The leading term is \(3x^3\) therefore the graph rises to the right and falls to the left based on the leading coefficient test.

Step 2: Finding the Zeros of the Polynomial

Setting the function equal to zero provides the x-intercepts of the graph. In this case, we solve for \(3x^3 - 24x^2=0\). First, factor out the greatest common factor, which is \(3x^2\), therefore, \(3x^2(x-8)=0\). Setting each factor equal to zero gives \(x=0\) and \(x=8\) as solutions.

Step 3: Plotting Sufficient Solution Points

Selected points are substituted into \(f(x)\) to create a table that maps \(x\) to \(f(x)\). Use a variety of points including those less than, equal to, and greater than the zeros. For example, consider (-1), (0), (1), (4), (8) and (9). Calculate \(f(x)=3x^3-24x^2\) for each selected \(x\).

Step 4: Drawing a Continuous Curve

Start by drawing the x and y axes on a graph and mark the x-intercepts (0, 0) and (8, 0). Calculate the value of the function at the selected test points and plot them. Draw a smooth continuous curve, guided by the plotted points and ensuring the curve passes through the plotted points. The end behavior of the curve should rise to the right and fall to the left.