Problem 40 - A Graphical Approach to College Algebra

Question

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=3 x^{4}-4 x^{3}-22 x^{2}+15 x+18 \\\&=(3 x+2)(x-3)\left(x^{2}+x-3\right)\end{aligned}$$

Step-by-Step Solution

Answer

Graph shows quartic curve with zeros at \(x = -\frac{2}{3}, 3, \frac{-1 \pm \sqrt{13}}{2}\), and turning points.

Step 1: Identify the degree of the polynomial

The polynomial given is of degree 4 because the highest power of \(x\) is 4. Thus, the graph will resemble a quartic function, which typically can have up to 3 turning points and begins and ends in the same direction (upward if the leading coefficient is positive).

Step 2: Identify the zeros of the polynomial

Look for the roots of the factors of the polynomial. From \((3x+2)=0\), we find \(x=-\frac{2}{3}\). From \(x-3=0\), we have \(x=3\). Solving \(x^2 + x - 3 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), which results in \(x = \frac{-1 \pm \sqrt{13}}{2}\). Hence, the zeros are \(x = -\frac{2}{3}\), \(x = 3\), \(x = \frac{-1 + \sqrt{13}}{2}\), and \(x = \frac{-1 - \sqrt{13}}{2}\).

Step 3: Determine the end behavior

Since the leading term is \(3x^4\) and its coefficient is positive, both the left and right ends of the graph will point upwards.

Step 4: Sketch the graph using zeros and end behavior

Plot the x-axis and mark the zeros found: \(x = -\frac{2}{3}\), \(x = 3\), \(x = \frac{-1 + \sqrt{13}}{2}\), and \(x = \frac{-1 - \sqrt{13}}{2}\). The graph starts from above the x-axis (because the end behavior is upwards), goes through each zero, and ends upwards again after \(x = 3\). Make sure the graph has turns between these points to represent up to 3 turns typical of a quartic polynomial.

Key Concepts

Polynomial Degree

The degree of a polynomial is determined by its highest power of the variable. In our exercise, the polynomial is given by \(P(x) = 3x^4 - 4x^3 - 22x^2 + 15x + 18\). The highest power of \(x\) here is 4, which means we are working with a quartic polynomial.
This defines several important characteristics of the graph:

• A quartic function can have up to 4 roots, or zeros, where it intersects the x-axis.
• Typically, this function will also have up to 3 turning points, where the direction of the graph changes.

This degree also shapes the basic framework of the graph's structure, especially when combined with the leading coefficient.

Roots of a Polynomial

The roots or zeros of a polynomial are the x-values for which the polynomial equals zero. Finding these roots is crucial for sketching the graph because each root corresponds to an x-intercept.
For our polynomial \((3x+2)(x-3)(x^{2}+x-3)\):

• The factor \((3x+2)\) gives the root \(x = -\frac{2}{3}\).
• The factor \((x-3)\) gives the root \(x = 3\).
• Using the quadratic formula on \(x^2 + x - 3\), we find two additional roots: \(x = \frac{-1 + \sqrt{13}}{2}\) and \(x = \frac{-1 - \sqrt{13}}{2}\).

These roots provide key points through which the graph will pass, helping define its shape and providing a guide for plotting.

End Behavior

End behavior describes how the polynomial behaves as \(x\) approaches infinity or negative infinity. This gives us an understanding of how the graph extends beyond the identifiable roots and turning points.
For our quartic polynomial, the term \(3x^4\) governs the end behavior. Since the leading coefficient (3) is positive, both ends of the graph will point upwards.

• As \(x\) approaches \(+\infty\), \(3x^4\) becomes larger and moves the graph upwards.
• As \(x\) approaches \(-\infty\), \(3x^4\) still dominates and the graph goes upward here as well.

This symmetric upward trend at both ends is typical for quartic functions with positive leading coefficients.

Graph Sketching

Graph sketching involves combining all the information we've gathered regarding the degree, roots, and end behavior to create a visual representation of the polynomial.
To sketch the polynomial from our exercise:

• Begin by plotting the x-axis and marking the roots \((-\frac{2}{3}, 0)\), \((3, 0)\), \((\frac{-1+\sqrt{13}}{2}, 0)\), and \((\frac{-1-\sqrt{13}}{2}, 0)\).
• Since the end behavior points both ends upward, sketch the graph entering from the top-left quadrant and exiting through the top-right quadrant.
• Ensure the graph has up to 3 turning points where the direction changes, consistent with the properties of a quartic function.

Each of these steps involves estimating where the graph slightly dips or rises between roots, giving a full shape consistent with the characteristics detailed earlier.