Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=2 x^{3}+9 x^{2}-6 x-40 \\ &=(x-2)(2 x+5)(x+4) \end{aligned}$$

Step-by-Step Solution

Answer

The function has roots at \(-4\), \(-\frac{5}{2}\), and \(2\), crosses the y-axis at \(-40\), and is cubic with up-right and down-left tails.

Step 1: Identify the Degree and Leading Term

The function is a polynomial of degree 3, as the highest power of x is 3. The leading term is \(2x^3\). Since this term has a positive coefficient, the graph rises to the right and falls to the left.

Step 2: Find the Roots of the Polynomial

From the factorized form \((x-2)(2x+5)(x+4)\), we can determine the roots: \(x=2\), \(x=-\frac{5}{2}\), and \(x=-4\). Each of these roots corresponds to an x-intercept on the graph.

Step 3: Determine the Y-intercept

To find the y-intercept, evaluate the function at \(x=0\): \(P(0) = (0-2)(2 \cdot 0+5)(0+4) = -2 \times 5 \times 4 = -40\). Thus, the y-intercept is at \((0, -40)\).

Step 4: Analyze the Behavior Around the Roots

Check the behavior of the polynomial at each root. Since all the factors are linear, the graph will cross the x-axis at each intercept. At \(x=2\), \(x=-\frac{5}{2}\), and \(x=-4\), the graph crosses the axis because the multiplicity of each root is 1.

Step 5: Sketch the Graph

Plot the points we have analyzed: the roots \((2, 0)\), \((-\frac{5}{2}, 0)\), \((-4, 0)\), and the y-intercept \((0, -40)\). Since the polynomial's leading term is positive, start by sketching a curve that descends from the left, crosses at \(x = -4\), \(x = -\frac{5}{2}\), then moves up to the right, crossing at \(x = 2\), and continues rising. Include the y-intercept at \((0, -40)\) in your sketch.

Key Concepts

Polynomial Roots

Polynomial roots are the solutions to the equation where the polynomial is equal to zero. For a third-degree polynomial like the one given, finding these roots helps us determine where the graph will intersect the x-axis. In this exercise, the polynomial is factorized as

• \((x-2)\)
• \((2x+5)\)
• \((x+4)\)

From these factors, we can easily extract the roots

• \(x = 2\)
• \(x = -\frac{5}{2}\)
• \(x = -4\)

These roots tell us exactly at which points the graph crosses or touches the x-axis. With this information, plotting the graph becomes a straightforward task. Remember, the sign of the coefficient for each squared term, if any, can affect the behavior at these roots, but in this case, all are linear, ensuring the graph crosses.

Y-intercept

The y-intercept of a polynomial graph is the point where the graph crosses the y-axis. This occurs when all other terms involving \(x\) are zero. In simpler terms, we find the y-intercept by evaluating the polynomial at \(x = 0\). To find the y-intercept of our polynomial function, substitute \(0\) into the equation: \[P(0) = (0-2)(2 \cdot 0 + 5)(0+4)\]Carrying out the calculations will give \[-2 \times 5 \times 4 = -40\]Thus, the y-intercept is at the point \((0, -40)\). Visualizing the graph, this point becomes crucial as it defines one of the extremities of the polynomial curve as it travels up or down the y-axis.

Graph Behavior

Understanding how a graph behaves at particular points is essential for sketching polynomial functions accurately. Let's explore how the graph behaves around its roots and other crucial points.When a root's factor appears only once, the graph crosses the x-axis at that root because it behaves like a straight line crossing through. In this case, each root's multiplicity is 1:

• At \(x = 2\), \(x = -\frac{5}{2}\), and \(x = -4\), the graph crosses.

Additionally, the nature of the leading term's coefficient affects how the graph behaves as \(x\) approaches positive and negative infinity.Ensure when sketching the graph that you're aware of these behaviors, which help determine the general curve and shape, aligning with the polynomial's degree.

Leading Term

The leading term of a polynomial function is the term with the highest power of \(x\). It's important because it primarily dictates the end behavior of the polynomial graph. For the given polynomial, the leading term is \(2x^3\). The coefficient (2) is positive, indicating that

• The graph will rise as \(x\) becomes very large (positive infinity).
• The graph will fall as \(x\) becomes very small (negative infinity).

This means that regardless of the number of roots or their multiplicities, the polynomial graph must rise on the right and fall on the left.Knowing this will help you sketch the graph more accurately, placing it well within the expected direction at any end of the x-axis. The leading term gives us significant insight into the 'big picture' feature of our graph's trajectory.