Problem 8 - Precalculus : Building Concepts and Connections

Question

Determine the multiplicities of the real zeros of the function. Comment on the behavior of the graph at the \(x\) -intercepts. Does the graph cross or just touch the \(x\) -axis? You may check your results with a graphing utility. $$g(s)=(s+6)^{4}(s-3)^{3}$$

Step-by-Step Solution

Answer

The function \(g(s) = (s+6)^{4}(s-3)^{3}\) has real zeros -6 and 3. The zero -6 has multiplicity 4 and the zero 3 has multiplicity 3. The graph of the function will touch the x-axis at \(s=-6\) and cross the x-axis at \(s=3\).

Step 1: Find the zeros

The zeros of the function are the solutions of the equation \(g(s) = 0\), so we have to find the values of \(s\) for which \((s+6)^{4}(s-3)^{3} = 0\). This happens when either \(s+6 = 0\) or \(s-3 = 0\). Solving these gives \(s = -6\) and \(s = 3\). Thus, the real zeros of the function are -6 and 3.

Step 2: Identify the multiplicities

In the given function \(g(s)=(s+6)^{4}(s-3)^{3}\), the factor \(s+6\) is raised to the exponent 4, so the corresponding zero, \(s = -6\), has a multiplicity of 4. Similarly, the factor \(s-3\) is raised to the exponent 3, so the corresponding zero, \(s = 3\), has a multiplicity of 3.

Step 3: Analyze the behavior of the graph at the zeros

A function crosses the x-axis at a zero if the zero has an odd multiplicity, and it just touches the x-axis without crossing if the zero has an even multiplicity. Therefore, for our function, at \(s=3\) the graph will cross the x-axis and at \(s=-6\) the graph will touch the x-axis without crossing.

Key Concepts

Multiplicities of Zeros

When we analyze a polynomial function, one vital concept that emerges is the multiplicity of a zero. Multiplicity refers to the number of times a particular zero appears as a factor in the polynomial. It's important to note that the multiplicity of a zero has critical implications for the graph of the function.

Let's take the polynomial function from our exercise, where the zeros are determined from \(g(s)=(s+6)^{4}(s-3)^{3}\). The zero \(s = -6\) is repeated four times, hence it has a multiplicity of 4, which is even. On the other hand, the zero \(s = 3\) is repeated three times, therefore it has a multiplicity of 3, which is odd.

Multiplicities give us insightful predictions about the behavior of the function's graph near the zero. For instance, at a zero with an even multiplicity, the graph will touch and rebound off the x-axis, while at a zero with an odd multiplicity, the graph will cross the x-axis. This distinct behavior helps us sketch the approximate shape of the graph without needing a precise plot.

Graph Behavior at X-Intercepts

Understanding how a polynomial graph behaves at its x-intercepts is crucial for visualizing the function's overall shape. As mentioned, the multiplicity of zeros plays a direct role in determining whether the graph crosses or merely touches the x-axis at these intercepts.

In the context of our exercise, the graph of the polynomial \(g(s)\) exhibits interesting behavior at its x-intercepts. For the zero with a multiplicity of 4 at \(s = -6\), we expect the graph to gently touch the x-axis and turn back on itself, creating a smooth, flattened curve at this intercept. This behavior is commonly referred to as 'bouncing off' the x-axis. Conversely, for the zero with multiplicity of 3 at \(s = 3\), the graph will cross the x-axis, creating a sharp point or a more distinct turn depending on the surrounding coefficients.

Graphing Utility Analysis

Graphing utilities, such as graphing calculators or software programs, are excellent tools for confirming our predictions about the behavior of polynomial functions. With these tools, we can effortlessly plot the function and visually validate the behavior at the zeros that we've deduced analytically.

When using a graphing utility to analyze \(g(s)\), we can observe the actual curvature and confirm our expectations: at \(s = -6\) the graph approaches the axis, touches it, and then bounces back due to the even multiplicity of 4. At \(s = 3\), the graph crosses the axis, reflective of the odd multiplicity of 3. This visual check reinforces our understanding and serves as practical affirmation of the algebraic analysis we've conducted on the polynomial's equation.

It's also worth mentioning that while graphing utilities are valuable for confirmation, developing the skill to predict graph behavior from the equation itself is an essential part of mathematical learning and offers students a deeper appreciation and insight into the nature of polynomial functions.