Problem 17 - 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. $$ P(x)=x(x-3)(x+2) $$

Step-by-Step Solution

Answer

The graph intercepts at \(x = -2, 0, 3\), with end behavior \(\uparrow \text{ as } x \to \infty\), and \(\downarrow \text{ as } x \to -\infty\).

Step 1: Identify the Roots

To sketch the polynomial function, start by finding its roots by setting \(P(x) = 0\). Solve \(x(x-3)(x+2) = 0\) to find the roots: \(x = 0\), \(x = 3\), and \(x = -2\). These are the x-intercepts of the graph.

Step 2: Determine the End Behavior

The end behavior of a polynomial function is determined by its leading term. Here, the highest degree term is \(x^3\) (since multiplying the factors would give \(x(x-3)(x+2) = x^3 + ...\)). Thus, as \(x\) approaches infinity, \(P(x)\) approaches infinity, and as \(x\) approaches negative infinity, \(P(x)\) approaches negative infinity.

Step 3: Identify the Y-Intercept

To find the y-intercept, evaluate the function at \(x=0\). Substituting \(x=0\) into the polynomial gives \(P(0)=0\). Hence, the y-intercept is \((0, 0)\).

Step 4: Sketch the Graph

Using the roots \(x = -2\), \(x = 0\), \(x = 3\) and the y-intercept at \((0,0)\), plot these points on a graph. Considering the end behavior, draw the curve such that it passes through the x-intercepts and moves upwards to the right and downwards to the left. The graph turns through each intercept since each factor is linear.

Key Concepts

Graph Sketching

Graph sketching is a useful skill when studying polynomial functions as it helps visualize the behavior of the function. When sketching, it's important to identify specific features like the roots and intercepts, as well as understanding the overall behavior of the graph.
Start by identifying the intercepts and critical points. Mark these points clearly on your graph. Then, take into account the polynomial's degree, which determines how the graph behaves as it extends towards positive and negative infinity.
Next, connect these points with a smooth curve. Make sure your graph reflects the correct end behavior. The graph should pass through all identified intercepts and turn according to the roots' multiplicity. Understanding these elements ensures that your sketch accurately represents the polynomial function's behavior.

Roots of a Polynomial

The roots of a polynomial, also known as zeros, are the values of \(x\) for which the polynomial equals zero. They are key in graph sketching because they give us the x-intercepts.
To find the roots, set the polynomial equal to zero and solve for \(x\). For example, with \(P(x) = x(x-3)(x+2)\), set each factor equal to zero: \(x=0\), \(x-3=0\), and \(x+2=0\). Solving these gives the roots \(x=0\), \(x=3\), and \(x=-2\).
The significance of roots in a graph is that they often indicate where the graph crosses the x-axis. In the polynomial \(P(x)\), since each root is of multiplicity one (appearing once), the graph will pass through the axis at a shallow angle.

End Behavior of Polynomials

The end behavior of a polynomial function describes how the graph behaves as \(x\) approaches infinity or negative infinity. This is largely determined by the degree of the polynomial and the leading coefficient.
The polynomial \(P(x) = x(x-3)(x+2)\) expands to \(x^3 + ...\), making it a cubic polynomial with a positive leading coefficient. This means as \(x\) tends towards infinity, \(P(x)\) increases, and as \(x\) tends towards negative infinity, \(P(x)\) decreases.
Understanding end behavior gives insight into the overall direction of the graph, allowing you to sketch it accurately. For this polynomial, expect the left end to swoop downwards, while the right end goes upwards, creating a unique curve.

X-Intercepts and Y-Intercepts

Intercepts are points where the graph crosses the axes. Calculating both x-intercepts and y-intercepts is vital for a complete sketch of a polynomial function.
An x-intercept occurs at the roots of the polynomial—that's where \(y=0\), and the graph crosses the x-axis. For \(P(x) = x(x-3)(x+2)\), we find the x-intercepts at \(x = 0, -2,\) and \(3\).
A y-intercept occurs when \(x=0\), and the value of \(P(x)\) at this point is \(P(0)=0\). Therefore, the graph crosses the y-axis at \((0,0)\). Both the x- and y-intercepts provide anchor points that help frame the shape and position of the graph on the coordinate plane.