Problem 19 - Algebra and Trigonometry

Question

\(15-26\) 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

The graph has x-intercepts at \(x = -2, \frac{2}{3}, 3\), y-intercept at 12, and rises on both ends.

Step 1: Identify the Degree and Leading Coefficient

The polynomial \( P(x) = (x-3)(x+2)(3x-2) \) can be expanded to find its degree. By multiplying the factors, the polynomial is: \( (x)(x)(3x) \), which gives \( 3x^3 \). Thus, the degree of the polynomial is 3, and the leading coefficient is 3.

Step 2: Determine the End Behavior

For a polynomial of degree 3 with a positive leading coefficient, the end behavior is such that as \( x \to +\infty \), \( P(x) \to +\infty \) and as \( x \to -\infty \), \( P(x) \to -\infty \).

Step 3: Find the Intercepts

To find the x-intercepts, set \( P(x) = 0 \):- \( x - 3 = 0 \) gives \( x = 3 \),- \( x + 2 = 0 \) gives \( x = -2 \),- \( 3x - 2 = 0 \) gives \( x = \frac{2}{3} \).Thus, the x-intercepts are \( x = 3 \), \( x = -2 \), and \( x = \frac{2}{3} \).The y-intercept is found by evaluating \( P(0) \):\( P(0) = (0 - 3)(0 + 2)(3 \times 0 - 2) = (-3)(2)(-2) = 12 \).So the y-intercept is 12.

Step 4: Sketch the Graph

Plot the x-intercepts at \( x = -2 \), \( x = \frac{2}{3} \), and \( x = 3 \) on the x-axis. Plot the y-intercept \( (0, 12) \) on the y-axis. Since the degree is odd, and the leading coefficient is positive, the graph will start in the third quadrant (where \( x \to -\infty \) and \( P(x) \to -\infty \)) and end in the first quadrant (where \( x \to +\infty \) and \( P(x) \to +\infty \)). Connect the intercepts with a smooth curve that reflects the polynomial's end behavior.

Key Concepts

Degree of Polynomial

The degree of a polynomial is a fundamental concept to understand. It tells you how many times the variable appears as a factor in the polynomial. In simple terms, it's the highest power of the variable when the polynomial is expanded and simplified. For the polynomial \( P(x) = (x-3)(x+2)(3x-2) \), we find the degree by considering the factors multiplying each other: \((x)(x)(3x)\). After simplifying, we get \(3x^3\). This shows that the polynomial is of degree 3. A polynomial of degree 3 is often called a cubic polynomial. Knowing the degree is important because it gives us insights into the graph's general shape and complexity. Generally, a degree of \(n\) can have up to \(n\) x-intercepts and at most \(n-1\) turning points. Understanding the degree helps in predicting how the graph might look.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In our example, after expanding to \(3x^3\), the leading coefficient is 3. It plays a crucial role in determining the behavior of the graph at the extremes (far left and far right). This is particularly true when discussing the polynomial's end behavior, which we will explore later. A positive leading coefficient, like in this example, generally indicates that the graph will rise to positive infinity on the right end for odd-degree polynomials. It is essential when analyzing how steeply the polynomial grows or shrinks and is key to comprehending the overall scale of the graph.

X-Intercepts

X-intercepts are points where the graph of the polynomial intersects the x-axis, essentially where the polynomial equals zero. They are vital for accurately sketching the graph. To find the x-intercepts, we set the polynomial \(P(x)\) to zero and solve for \(x\). From \(P(x)=(x-3)(x+2)(3x-2)\), we get:

• \(x - 3 = 0\), resulting in \(x = 3\)
• \(x + 2 = 0\), which gives \(x = -2\)
• \(3x - 2 = 0\), leading to \(x = \frac{2}{3}\)

These solutions correspond to the x-intercepts at \(x = 3\), \(x = -2\), and \(x = \frac{2}{3}\). X-intercepts tell us where the graph crosses the x-axis, and understanding this aids in constructing the graph's accurate shape.

End Behavior

End behavior describes what happens to the graph of a polynomial function as \(x\) approaches positive or negative infinity. It's influenced by the degree and the leading coefficient. For the cubic polynomial \(P(x) = 3x^3\) with a positive leading coefficient, the end behavior can be concluded as follows:

• As \(x \to +\infty\), \(P(x) \to +\infty\)
• As \(x \to -\infty\), \(P(x) \to -\infty\)

In simple terms, it means that as you move towards the right on the x-axis, the graph goes upwards, and as you move towards the left, the graph goes downwards. Recognizing the end behavior helps with sketching the beginning and ending behavior of the polynomial graph, ensuring it mirrors the reality of the function's nature.