Question

Find polynomials \(p\) and \(q\) such that \(p / q\) is undefined at 1 and \(2,\) but \(p / q\) has a vertical asymptote only at \(2 .\) Sketch a graph of your function.

Short answer

Question: Find the polynomials p and q such that p/q is undefined at 1 and 2, but has a vertical asymptote only at 2. Answer: For p/q to be undefined at x=1 and x=2, both p and q must have a factor of (x-1). For there to be a vertical asymptote only at x=2, q must have an additional factor of (x-2). Therefore, we can build the polynomials as follows: p(x) = (x - 1) P(x) q(x) = (x - 1)(x - 2) Q(x) where P(x) and Q(x) are arbitrary polynomials that do not share any common factors.

Step-by-step solution

Building the polynomials

Following the analysis, let's build the polynomials for which the ratio \(p/q\) will have the desired properties. Let $$p(x) = (x - 1) P(x)$$ and $$q(x) = (x - 1)(x - 2) Q(x),$$where \(P(x)\) and \(Q(x)\) are two arbitrary polynomials that don't share any common factors. This construction guarantees that both \(p\) and \(q\) will be undefined at 1 and 2, and the function will have a vertical asymptote at 2.

Analyzing the behavior of the function

Now let's analyze the behavior of the function $$f(x) = \frac{p(x)}{q(x)} = \frac{(x - 1) P(x)}{(x - 1)(x - 2) Q(x)}.$$ In the neighborhood of 1, we can cancel the common factor \((x - 1)\): $$\lim_{x \rightarrow 1} \frac{(x - 1) P(x)}{(x - 1)(x - 2) Q(x)} = \lim_{x \rightarrow 1} \frac{P(x)}{(x - 2) Q(x)}.$$ As the limit exists, there is no vertical asymptote at 1. Similarly, we can analyze the behavior of the function near 2 by finding the limit as \(x\) approaches 2: $$\lim_{x \rightarrow 2} \frac{(x - 1) P(x)}{(x - 1)(x - 2) Q(x)} = \lim_{x \rightarrow 2} \frac{P(x)}{(x - 2) Q(x)}.$$ Since the denominator approaches 0 and the numerator does not approach 0, the limit does not exist and there is a vertical asymptote at 2.

Sketching the graph

To sketch the graph, we can use the properties derived in the previous steps: 1. The function is undefined at 1 and 2. 2. There is a vertical asymptote at 2. Using these facts, we can make a basic sketch of the graph, ensuring that there are vertical asymptotes at x=2 and holes at x=1. The exact shape of the graph will depend on the specific choice of polynomials \(P(x)\) and \(Q(x)\). However, the important points related to the vertical asymptote and the undefined points will remain the same as shown in the sketch.

Key concepts

Vertical Asymptotes

Vertical asymptotes are crucial features of the graphs of rational functions. They are essentially the x-values at which a function diverges to infinity or negative infinity, as it does not exist at these points. In the context of rational functions, vertical asymptotes occur where the denominator equals zero and the numerator does not zero out these terms.

For our function, the vertical asymptote is found at the x-value of 2. This is because, when we substitute x = 2 into the factor \(x - 2\) in the denominator, the function becomes undefined due to division by zero. Meanwhile, cancelling common factors like \(x - 1\) between the numerator and denominator prevents a vertical asymptote at x = 1.

The important takeaway is the role of cancelling factors. Factors in both the numerator and the denominator that are identical can be simplified, typically leaving a hole in the graph rather than a vertical asymptote. However, where the denominator becomes zero exclusively, it leads to a vertical asymptote at that point.

Undefined Points

Understanding undefined points in rational functions helps clarify where the function cannot exist. These points arise when the denominator equals zero. Such places are where the function potentially stops behaving in a typical manner and instead, may head towards infinity or create gaps in the graph.

In reference to the function \(f(x) = \frac{p(x)}{q(x)}\), both x = 1 and x = 2 are initially considered undefined due to these being roots of the denominator. However, upon canceling the \(x-1\) factor, the function becomes defined at x = 1. This conversion from an undefined point to a hole shows how simplification can change the nature of these points.

A crucial skill is determining which points are truly undefined and which are merely holes upon simplifying the function. Identifying these distinctions helps understand a function's domain and asymptotic behavior, influencing the graph's depiction.

Polynomial Division

Polynomial division is a technique used to divide polynomials similarly to numerical division. In the realm of rational functions, it helps simplify expressions by reducing common factors from the numerator and the denominator. This process can greatly influence the presence or absence of undefined points and vertical asymptotes.

Let's take \(p(x) = (x - 1)P(x)\) and \(q(x) = (x - 1)(x - 2)Q(x)\). Here, both the numerator and the denominator have \(x - 1\) as a common factor. Polynomial division allows us to eliminate the common factor, implying that rather than experiencing a true division by zero error at x = 1 (an undefined point), the point becomes a hole.

Simplifying expressions in this way often makes them easier to evaluate and graph. It helps reveal the fundamental structure of the rational function, distinguishing between different types of discontinuities like holes and asymptotes in a clearer manner.