Problem 15 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Plot the x-intercepts \(x = -2, \frac{2}{3}, 3\), the y-intercept \((0,12)\), and show the end behavior.

Step 1: Identify the factors and roots

The polynomial function given is \( P(x) = (x-3)(x+2)(3x-2) \). To find the roots, set each factor equal to zero: \( x-3=0 \), \( x+2=0 \), and \( 3x-2=0 \). This gives us the roots \( x = 3 \), \( x = -2 \), and \( x = \frac{2}{3} \). These are the x-intercepts of the graph.

Step 2: Determine the y-intercept

To find the y-intercept, substitute \( x = 0 \) in the polynomial function. \( P(0) = (0-3)(0+2)(3 \times 0 - 2) = (-3)(2)(-2) = 12 \). Thus, the y-intercept is the point \( (0, 12) \).

Step 3: Analyze the end behavior

The end behavior is determined by the leading term when the polynomial is expanded. The leading term is obtained by multiplying the leading terms of each factor: \( x \cdot x \cdot 3x = 3x^3 \). Since the leading term is \( 3x^3 \), a positive cubic, the end behavior is: as \( x \to \infty \), \( P(x) \to \infty \) and as \( x \to -\infty \), \( P(x) \to -\infty \).

Step 4: Sketch the graph

Plot the intercepts: the roots \((3,0)\), \((-2,0)\), and \((\frac{2}{3},0)\) and the y-intercept \((0,12)\). Use the end behavior to sketch the curve: starting from below the x-axis moving upward through \(-2\), reaching above the x-axis at the y-intercept, crossing back through \(\frac{2}{3}\) and continuing in a downward trend, then curving upward through \(3\). The graph will have a general 'S' shape consistent with cubic functions.

Key Concepts

Roots of Polynomial

In polynomial graphing, identifying the roots of the polynomial is an essential first step. Roots of a polynomial are the x-values where the polynomial intersects the x-axis, also known as the x-intercepts. For a function like \(P(x) = (x-3)(x+2)(3x-2)\), we can determine the roots by setting each factor equal to zero. Here's how it breaks down:

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

Once these values are calculated, you have the roots \(x = 3\), \(x = -2\), and \(x = \frac{2}{3}\). These points tell us where the polynomial will cross the x-axis. Understanding where these roots are located helps in sketching the overall graph and predicting its shape.

End Behavior of Polynomials

The end behavior of a polynomial describes how the graph behaves as \(x\) approaches infinity or negative infinity. This is crucial for understanding the overall direction and shape of the graph. In our case, we focus on the polynomial \(3x^3\) because it is the term with the highest degree and defines the polynomial's leading behavior.

• A positive leading coefficient with an odd degree, like \(3x^3\), means as \(x \to \infty\), \(P(x) \to \infty\) and as \(x \to -\infty\), \(P(x) \to -\infty\).

Therefore, the graph of \(P(x)\) will rise to infinity on the right and fall to negative infinity on the left, creating a characteristic shape that is often seen in cubic functions. Recognizing the end behavior allows you to sketch the tails of the graph accurately.

Intercepts of Graphs

Intercepts are crucial points on a polynomial graph, indicating where the graph crosses the axes. These intersections include both x-intercepts and the y-intercept. For our polynomial function \((x-3)(x+2)(3x-2)\), we already know the x-intercepts:

• \(x = 3\)
• \(x = -2\)
• \(x = \frac{2}{3}\)

To find the y-intercept, substitute \(x = 0\) into the function:\[P(0) = (0-3)(0+2)(3 \times 0 - 2) = (-3)(2)(-2) = 12\]Thus, the y-intercept is at \((0, 12)\). Knowing both the x- and y-intercepts allows you to anchor the main framework of the graph, ensuring accurate depiction of the polynomial's path on the coordinate planes.

Cubic Functions

Cubic functions are polynomials of degree three, taking the general form \(ax^3 + bx^2 + cx + d\). These functions have a distinctive 'S' shaped curve, due to their end behavior which is influenced primarily by their highest degree term. For example, consider the polynomial function \(P(x) = (x-3)(x+2)(3x-2)\). After expanding, the leading term is \(3x^3\).

• This leading term suggests a classic cubic shape: crossing the x-axis three times at its roots.

Cubic functions can have inflection points where the curvature changes direction, adding complexity to their graphs. This inflection is visible as the graph moves through its intercepts and creates the snake-like pattern observed in cubic curves. When sketching cubic functions, it's crucial to apply all known intercepts and their end behavior, ensuring that the graph accurately reflects these mathematical properties.