Problem 17 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Zeros: \(x = 1\) (touches), \(x = 3\) (crosses); Y-intercept: \(-3\). End behavior: \(-\infty\ to +\infty\).

Step 1: Identify the Zeros

The zeros of the polynomial function can be found by setting the equation equal to zero: \(P(x) = (x-1)^{2}(x-3) = 0\). This gives the solutions \(x = 1\) with multiplicity 2, and \(x = 3\).

Step 2: Determine Y-Intercept

To find the y-intercept, substitute \(x = 0\) into the polynomial: \(P(0) = (0-1)^{2}(0-3) = 1 \times (-3) = -3\). Thus, the y-intercept is at (0, -3).

Step 3: Analyze Multiplicities for Behavior at Zeros

At \(x = 1\), the zero has multiplicity 2, indicating a parabola-like behavior, where the graph touches the x-axis but does not cross it. At \(x = 3\), the multiplicity is 1, indicating the graph will cross the x-axis.

Step 4: Check End Behavior

Since the degree of \(P(x)\) is 3 (odd), with a positive leading coefficient, the end behavior of the graph is: as \(x \to -\infty\), \(P(x) \to -\infty\), and as \(x \to \infty\), \(P(x) \to \infty\).

Step 5: Sketch the Graph

Start by plotting the points at the zeros \(x = 1\) and \(x = 3\), as well as the y-intercept \(y = -3\). The graph will touch the x-axis at \(x=1\) and cross the x-axis at \(x=3\), while following the identified end behavior.

Key Concepts

Zeros of Polynomials

The zeros of a polynomial are the x-values where the polynomial equation equals zero. In layman's terms, these are the points where the graph will either touch or cross the x-axis. For the polynomial function given, \( P(x) = (x-1)^2(x-3) \), the zeros can be found by setting each factor equal to zero. This results in solutions \( x = 1 \) and \( x = 3 \). These zeros have particular characteristics due to their multiplicities:

• **Multiplicity**: At \( x = 1 \), the zero has a multiplicity of 2. This means the graph will touch the x-axis but not cross it, resembling a "bounce" effect similar to a parabola at this point.
• **Simple Zero**: At \( x = 3 \), the zero has a multiplicity of 1, which indicates that the graph will cross the x-axis at this point.

Understanding these properties is crucial because they dictate how the graph behaves at each zero. It helps in predicting where the graph will change direction.

Intercepts of Graphs

Intercepts are points where the graph intersects the axes. These include both x-intercepts (already discussed as zeros) and the y-intercept.

• The **y-intercept** is found by setting \( x = 0 \) and solving for \( y \). For our polynomial, we substitute \( x = 0 \):

\[P(0) = (0-1)^2(0-3) = 1 \times (-3) = -3\]So, the y-intercept is \((0, -3)\). This point provides a starting reference for sketching the graph. Each intercept provides a checkpoint on the graph, helping to define the framework of the curve. By connecting these points with the expected behavior, the overall shape of the graph can be sketched more accurately.

Graph End Behavior

End behavior describes how the graph of a polynomial function behaves as \( x \) approaches infinity or negative infinity. This behavior is mainly determined by the degree and the leading coefficient of the polynomial.In our function \( P(x) = (x-1)^2(x-3) \), the degree is 3, an odd number, with a positive leading coefficient. Therefore, the end behavior follows these rules:

• As \( x \to \infty \), \( P(x) \to \infty \): The graph rises to the right.
• As \( x \to -\infty \), \( P(x) \to -\infty \): The graph falls to the left.

This knowledge is vital because it frames the overall shape of the graph. By understanding end behavior, students can predict how the graph extends beyond the intercepts, ensuring the sketch aligns with the actual mathematical properties of the polynomial.