Problem 35 - 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.) $$\begin{aligned}P(x) &=2 x^{4}+x^{3}-6 x^{2}-7 x-2 \\\&=(2 x+1)(x-2)(x+1)^{2}\end{aligned}$$

Step-by-Step Solution

Answer

The polynomial has zeros at \(-\frac{1}{2}\), \(-1\) (bounce), and \(2\); the y-intercept is \(-2\); and the ends of the graph go upwards.

Step 1: Identify the degree and leading coefficient

The polynomial given is a 4th degree polynomial with a leading term of \(2x^4\). The leading coefficient is 2, which is positive.

Step 2: Determine the end behavior

Since the degree of the polynomial is even and the leading coefficient is positive, as \(x\) approaches \(\pm \infty\), the function \(P(x)\) approaches positive infinity. This means the graph's tails will both point upwards.

Step 3: Find the zeros of the polynomial

From the factorized form \((2x+1)(x-2)(x+1)^{2}\), we identify the zeros: \(x = -\frac{1}{2}\), \(x = 2\), and \(x = -1\), with \(x = -1\) having a multiplicity of 2, indicating a bounce at this root.

Step 4: Analyze the behavior at each zero

- At \(x = -\frac{1}{2}\), the graph crosses the x-axis.- At \(x = 2\), the graph also crosses the x-axis.- At \(x = -1\), the graph touches the x-axis and bounces back.

Step 5: Determine y-intercept

Substitute \(x = 0\) into the polynomial: \[ P(0) = (2 \times 0 + 1)(0 - 2)(0 + 1)^{2} = (1)(-2)(1) = -2 \]. The y-intercept is \( (0, -2) \).

Step 6: Sketch the graph

- Start at the y-intercept \((0, -2)\).- The graph bounces off the x-axis at \(x = -1\).- It crosses the x-axis at \(x = -\frac{1}{2}\) and \(x = 2\).- The ends of the graph should point upwards, due to the even degree and positive leading coefficient.

Key Concepts

End Behavior

Understanding the end behavior of a polynomial graph is essential. The "end behavior" refers to how the graph of a polynomial behaves as the input values (the x-values) grow larger in positive or negative direction. Here, a few key factors determine this behavior:

• The degree of the polynomial (even or odd).
• The sign of the leading coefficient (positive or negative).

For the polynomial \( P(x) = 2x^4 + x^3 - 6x^2 - 7x - 2 \), the degree is 4 (even) and the leading coefficient is 2 (positive). This tells us as \( x \to \pm \infty \), \( P(x) \to \infty \). This means both ends of the graph will point upwards.

Zeros of Polynomials

The zeros of a polynomial are the x-values where the polynomial equals zero. These are crucial since they indicate where the graph intercects or touches the x-axis. In our polynomial, the factored form \( (2x+1)(x-2)(x+1)^2 \) already reveals the zeros at:

• \( x = -\frac{1}{2} \) from \( 2x + 1 \)
• \( x = 2 \) from \( x - 2 \)
• \( x = -1 \) from \( (x+1)^2 \)

Finding the zeros helps us plot where the graph will either cross or bounce off the x-axis. Each of these points is a guidepost during graph sketching.

Multiplicity

When discussing the zeros of polynomials, it's important to recognize the multiplicity of each zero, which tells us about the behavior of the graph at that zero. In simpler terms, multiplicity refers to how many times a particular zero appears in the factorization. For example:

• A zero with an odd multiplicity (like 1) causes the graph to cross the x-axis.
• A zero with an even multiplicity (like 2) causes the graph to "bounce" off the x-axis at that point.

In \( P(x) = (2x+1)(x-2)(x+1)^2 \), the multiplicity of the zero at \( x = -1 \) is 2, meaning the graph will touch the x-axis and bounce back up at this point. Such insights help provide a clearer picture of the graph's behavior.

Y-Intercept

The y-intercept is the point where the graph intersects the y-axis. This happens when \( x = 0 \). To find it, simply substitute \( x = 0 \) into the polynomial equation and solve for \( P(0) \). For our polynomial:\[P(0) = (2 \times 0 + 1) (0 - 2) (0 + 1)^2 = -2\]So, the y-intercept is \((0, -2)\). This is a specific point that marks the graph's starting point along the vertical axis and serves as a anchor for sketching the graph accurately. Knowing the y-intercept provides another crucial reference point when drawing the graph by hand or analyzing it further.