Problem 85 - 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. \(f(x)=x^{3}-9 x\)

Step-by-Step Solution

Answer

The graph of the function \(f(x) = x^{3} - 9x\) passes through the points \((-3, 0)\), \((0, 0)\), and \((3, 0)\) and falls to the left and rises to the right. More accuracy for the graph can be achieved by plotting a few more points - particularly between zeros and on either side of extreme zeros.

Step 1: Apply the Leading Coefficient Test

Analyze the leading coefficient (the highest power's coefficient) in the equation. Here, the leading coefficient is 1 associated with \(x^{3}\). The leading coefficient test states if the leading term's coefficient is positive and the degree of the function is odd, then the function rises to the right and falls to the left. Hence, in this case, the function falls to the left and rises to the right.

Step 2: Find the Zeros of the Polynomial

Setting the equation equal to zero and solving for x will provide the zeros of the polynomial, i.e., \(x^{3}-9x = 0\). Here, take x common to end up with x*(x^2 - 9) = 0. This gives us three solutions: \(x=0\), \(x=3\) and \(x=-3\). These are the x-intercepts of the graph.

Step 3: Plot Sufficient Solution Points

For a cubic equation, having the zeros and leading coefficient information is generally sufficient to sketch a decent graph. However, for more accuracy plot additional points, particularly between zeroes and on either side of extreme zeroes. Consider points \(x=-4\), \(x=-2\), \(x=2\), \(x=4\). To get the corresponding y values, just substitute each chosen value for \(x\) into the function \(f(x)\).

Step 4: Draw a Continuous Curve through the Points

The last step is to take the points from step 2 and 3 and plot them on a graphical axis. Then, draw a smooth, continuous curve that passes through all of these points. Remember that the curve falls to the left and rises to the right according to the leading coefficient test.

Key Concepts

Cubic Functions

A cubic function is a type of polynomial function that is characterized by having the highest exponent of 3. It is commonly written in the form \(f(x) = ax^3 + bx^2 + cx + d\), where \(a, b, c,\) and \(d\) are constants with \(a eq 0\). This type of function can model various real-world phenomena, from physics to economics.

Cubic functions have certain key features:

• Three roots or zeros, which are the values of \(x\) that make \(f(x) = 0\).
• One or two turning points, where the function changes direction.
• Can have up to three intersections with the x-axis, depending on the nature of its roots.

Understanding the shape of a cubic function is crucial, as it can rise and fall in different ways depending on its coefficients. The behavior at the extremes, that is for very large or small values of \(x\), is determined by the leading term \(ax^3\). Analyzing this allows for accurate graph sketching.

Leading Coefficient Test

The Leading Coefficient Test is a handy tool for determining the end behavior of a polynomial function based on its leading term. The leading term is the term with the highest power, which dominates the behavior of the polynomial as \(x\) approaches infinity or negative infinity.

For cubic functions, the leading coefficient test can be summarized as:

• If the leading coefficient is positive and the degree is odd, such as in the function \(f(x) = x^3 - 9x\), the graph will generally rise to the right and fall to the left.
• If the leading coefficient is negative and the degree is odd, the graph will generally fall to the right and rise to the left.

This test is useful for quickly sketching the general shape of the function by focusing on its behavior as \(x\) tends towards positive and negative infinity. It provides a helpful guideline for understanding how the graph behaves outside of its critical points.

Zeros of a Polynomial

The zeros of a polynomial function are the values of \(x\) where \(f(x) = 0\). Finding these zeros is crucial because they tell us where the graph of the function intersects the x-axis.

To find the zeros of a cubic polynomial like \(f(x) = x^3 - 9x\), you first set the function equal to zero and solve for \(x\). Here, factoring out the common factor gives \(x(x^2 - 9) = 0\). This further breaks down to \((x)(x-3)(x+3) = 0\), so the solutions or zeros are \(x=0\), \(x=3\), and \(x=-3\).

Knowing these zeros:

• Helps in plotting key points on the graph.
• Identifies intercepts with the x-axis.

Determining zeros is an essential step in sketching polynomial graphs accurately, as they divide the graph into distinct sections of behavior.

Plotting Points

Plotting points is the process of evaluating the function at various values of \(x\) and marking these coordinates on a graph. This helps in visualizing the graph with more precision.

For the function \(f(x) = x^3 - 9x\), we start by plotting zeros: \(x=0, x=3,\) and \(x=-3\). This gives the points \((0,0), (3,0),\) and \((-3,0)\).

To gain a fuller picture, calculate additional points by choosing \(x\) values around and between these zeros, like \(-4, -2, 2,\) and \(4\). Substitute these back into the function to find corresponding \(y\) values:

• \(f(-4) = (-4)^3 - 9(-4) = -64 + 36 = -28\)
• \(f(-2) = (-2)^3 - 9(-2) = -8 + 18 = 10\)
• \(f(2) = (2)^3 - 9(2) = 8 - 18 = -10\)
• \(f(4) = (4)^3 - 9(4) = 64 - 36 = 28\)

Drawing a smooth curve through these plotted points enables the visualization of the cubic function graph. Each plotted point ensures the graph accurately reflects the function's behavior.