Problem 14 - Algebra and Trigonometry

Question

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=(2 x-1)(x+1)(x+3) $$

Step-by-Step Solution

Answer

Draw the graph with x-intercepts at \(-3, -1, \frac{1}{2}\), a y-intercept at \(-3\), and end behavior rising to the right and falling to the left.

Step 1: Identify the roots of the polynomial

The polynomial is given in factored form: \( P(x) = (2x-1)(x+1)(x+3) \). We can find the roots by setting each factor equal to zero. This gives roots at \( x = \frac{1}{2}, x = -1, \text{ and } x = -3 \). These are the x-intercepts of the graph.

Step 2: Determine the y-intercept

To find the y-intercept, substitute \( x = 0 \) into the polynomial. \[ P(0) = (2(0) - 1)(0 + 1)(0 + 3) = (-1)(1)(3) = -3 \]. So, the y-intercept is \( (0, -3) \).

Step 3: Analyze the end behavior

The polynomial \( P(x) = (2x-1)(x+1)(x+3) \) has a degree of 3, since there are three linear factors. The leading term is \( 2x^3 \). As \( x \to \infty \), \( P(x) \to \, \infty \) because a positive cubic term increases. As \( x \to -\infty \), \( P(x) \to \, -\infty \), because the leading term remains negative for large negative \( x \).

Step 4: Sketch the graph

Using the roots \( x = \frac{1}{2}, x = -1, \text{ and } x = -3 \) and the y-intercept \( (0, -3) \), sketch the graph starting from the bottom right (since it rises to the right) and passing through these points. The curve should start from \( -\infty \) as \( x \to -\infty \), rise through \( x = -3 \), dip through \( x = -1 \), rise again through \( x = \frac{1}{2} \), and continue upward as \( x \to \infty \).

Key Concepts

Polynomial Roots

The roots of a polynomial are the values of \(x\) where the polynomial equals zero. These are also known as the x-intercepts of the graph because they occur at points where the graph crosses the x-axis. In the case of our polynomial \(P(x) = (2x-1)(x+1)(x+3)\), setting each factor equal to zero allows us to easily find these roots. This gives us roots at \(x = \frac{1}{2}\), \(x = -1\), and \(x = -3\).
To find the roots, consider solving these simple equations:

• For \(2x-1=0\), solve to get \(x = \frac{1}{2}\).
• For \(x+1=0\), solve to get \(x = -1\).
• For \(x+3=0\), solve to get \(x = -3\).

At these x-values, the polynomial graph will touch or cross the x-axis.

Intercepts

Intercepts play a crucial role in the graph of a polynomial function, signaling the points where the graph meets the axes. Besides the x-intercepts (roots), another key intercept is the y-intercept. This is the point where the graph crosses the y-axis. To find the y-intercept of a polynomial, we evaluate the polynomial at \(x = 0\).
Substituting \(x = 0\) in \(P(x)\):
\[ P(0) = (2 \times 0 - 1)(0 + 1)(0 + 3) = -1 \times 1 \times 3 = -3 \]
This calculation shows the y-intercept is \((0, -3)\).
Thus, our polynomial graph will cross the y-axis at the point \((0, -3)\). Remember, intercepts easily guide you towards sketching an accurate graph.

End Behavior

The end behavior of a graph describes how the graph acts as \(x\) approaches infinity or negative infinity. For any polynomial, this is determined by its leading term when fully expanded.
In \(P(x) = (2x-1)(x+1)(x+3)\), the degree of the polynomial is 3, because there are three linear factors. The leading term after multiplying these factors is \(2x^3\).
Following these guidelines:

• If the leading coefficient is positive and the degree is odd, like our \(2x^3\), the graph rises to the right and falls to the left.
• As \(x \to \infty\), \(2x^3 \to \infty\)
• As \(x \to -\infty\), \(2x^3 \to -\infty\)

This pattern is what guides the overall shape and direction of the polynomial graph.

Factored Form

Understanding the factored form of a polynomial is crucial as it provides direct insight into the roots of the polynomial. Factored form is an expression where the polynomial is written as a product of its linear factors.
For \(P(x) = (2x-1)(x+1)(x+3)\), we already have it in factored form. Each factor \((2x-1)\), \((x+1)\), and \((x+3)\) represents a simpler expression we can solve for to find the polynomial's roots.
Factored form offers these benefits:

• Immediate identification of x-intercepts at solutions of each factor (discussed in polynomial roots).
• Helpful for sketching polynomial graphs, as knowing where roots and intercepts are is crucial.
• It simplifies analyzing the polynomial without the need to perform lengthy algebraic expansions.

Remember, writing polynomials in factored form makes graphing, evaluating, and understanding their behavior much easier.