Problem 109 - Algebra and Trigonometry

Question

Touches the x-axis at 0 and crosses the x-axis at 2; lies below the x-axis between 0 and 2

Step-by-Step Solution

Answer

The function that meets the given conditions is \(f(x) = -x^{2}(x - 2)\)

Step 1: Identifying the Roots

A function touches the x-axis at a point (indicative of a root) without crossing over if that root is of even multiplicity, while it crosses the x-axis at a point if that root is of odd multiplicity. Hence, we can deduce that the function has a root of 0 of at least multiplicity 2 (let's assume 2 for simplicity) and a root of 2 of multiplicity 1.

Step 2: Formulating the Function

A polynomial function can be written in factored form as \(f(x) = a(x - r1)^{m1}(x - r2)^{m2}...\) where \(a\) is a constant (which can be assumed to be 1 for simplicity), \(r1, r2, ...\), are the roots of the function, and \(m1, m2, ... \) are corresponding multiplicities of the roots. Given the roots and their multiplicities from step 1, we can write the function as \(f(x) = 1*(x - 0)^{2}*(x - 2)^{1}\).

Step 3: Adjust for Negative Region

The function given in step 2 will cross x-axis at 0 and 2, but it will lie above the x-axis between 0 and 2. We want it to be below the x-axis between 0 and 2. Therefore we introduce a negative sign to flip the function. This gives the final function as: \(f(x) = -1*(x - 0)^{2}*(x - 2)^{1}\), or simplifying gives: \(f(x) = -x^{2}(x - 2)\)

Key Concepts

Multiplicity of Roots

Understanding the multiplicity of roots is crucial when analyzing polynomial functions. The multiplicity of a root refers to the number of times a particular root is repeated in the polynomial.

• If a polynomial has a root of multiplicity 1, the graph will cross the x-axis at that point.
• For a root with an even multiplicity, the graph touches the x-axis but does not cross it.
• An odd multiplicity, that is greater than 1, means the graph will cross the x-axis in a different manner, appearing flatter and more gentle at that point.

In the original problem, the root at 0 has a multiplicity of 2, indicating the graph touches the x-axis at that point without crossing. Whereas, the root at 2 has a multiplicity of 1, revealing that the graph crosses the x-axis there.

Factored Form of Polynomial

The factored form of a polynomial function allows us to easily identify roots and their multiplicities. A polynomial in this form is represented as:\[f(x) = a(x - r1)^{m1}(x - r2)^{m2}...\]

• Here, \(a\) is a coefficient that can stretch or compress the graph, often set to 1 initially to simplify analysis.
• \((x - ri)\) is a factor corresponding to root \(ri\), and \(mi\) represents the multiplicity of that root.

The exercise gives us the polynomial function with roots 0 and 2. Their multiplicities are 2 and 1, respectively. This results in the function:\[f(x) = 1 \cdot (x - 0)^{2} \cdot (x - 2)^{1}\]Factoring polynomials makes it easier to predict the overall shape of the graph by looking at the behavior around its roots.

Roots of Polynomial Functions

Roots, also known as zeroes of a polynomial function, are the values at which the function equals zero, as the graph intersects the x-axis. Finding these roots is essential for understanding and sketching the polynomial's graph.

• The location and multiplicity of the roots determine the behavior of the graph around those points.
• Even multiplicity roots cause the graph to "bounce" off the x-axis, whereas odd multiplicity roots allow the graph to "cut through" the axis.

In the given problem, we identify roots at 0 and 2. The root at 0 (even multiplicity) causes the graph to briefly touch the x-axis, while the root at 2 signifies the graph will cross the x-axis.Additionally, adjusting the function such as introducing a negative sign, can affect the region of the graph, flipping it across the x-axis as required to ensure its behavior in the negative region between two roots. Our final function from the exercise \(f(x) = -x^{2}(x - 2)\) ensures that the graph lies below the x-axis between the intervals of those roots.