Problem 108 - College Algebra Essentials

Question

Graph \(y_{1}=x^{2}-2 x, y_{2}=x,\) and \(y_{3}=y_{1} \div y_{2}\) in the same \([-10,10,1]\) by \([-10,10,1]\) viewing rectangle. Then use the TRACE feature to trace along \(y_{3} .\) What happens at \(x=0 ?\) Explain why this occurs.

Step-by-Step Solution

Answer

At \(x=0\), the trace on \(y_{3}\) goes to negative or positive infinity depending upon from which side you approach zero. This happens because at \(x=0\), the function \(y_{3}\) involves division by zero which is undefined, hence the trace tends to approach negative or positive infinity.

Step 1: Plotting \(y_{1}\) and \(y_{2}\)

By using the mentioned viewing rectangle, first plot \(y_{1}=x^{2}-2 x\) and \(y_{2}=x\) on the graph. Use your function to plot.

Step 2: Plotting \(y_{3}\)

Now, create the plot of \(y_{3}\) by dividing values of \(y_{1}\) by corresponding values of \(y_{2}\) at each x. This will be the function \(y_{3}=y_{1} / y_{2}\). Plot this on the graph.

Step 3: Use the TRACE feature

Using the TRACE feature, follow the function \(y_{3}\) and track its behavior as \(x\) approaches 0. Document your observations.

Step 4: Observations at \(x=0\)

You will notice that at \(x=0\), irrespective of whether you approach it from the positive side or the negative side, the trace on \(y_{3}\) will reach infinity or negative infinity, respectively. This is because you are trying to divide by 0 at \(x=0\) (since \(y_{2}=x\)), and division by zero is undefined.

Step 5: Explanation of the phenomenon

This situation arises due to the nature of division by zero. As \(y_{2}=x\) approaches 0, you are essentially dividing by smaller and smaller quantities in \(y_{3}=y_{1} / y_{2}\), causing the result to approach infinity or negative infinity depending on the sign of the quantity divided. This is why the trace of \(y_{3}\) shoots to these values as \(x\) approaches 0.

Key Concepts

Asymptotes in Algebra

An asymptote is a line that a graph approaches but never touches or crosses. Asymptotes can be horizontal, vertical, or oblique, and they provide vital information about the behavior of a graph as the inputs increase towards positive or negative infinity.

In algebra, vertical asymptotes are found where the function cannot have a value, which typically happens when there's a division by zero. Take for example the function discussed in the exercise:
\(y_{3}=\frac{y_{1}}{y_{2}}=\frac{x^{2}-2x}{x}\). As \(x\) approaches zero, the denominator of the rational expression becomes smaller and smaller, forcing the value of \(y_{3}\) to grow without bounds, illustrating that there is a vertical asymptote at \(x=0\).

Horizontal asymptotes, on the other hand, indicate the value that the function approaches as \(x\) grows large in the positive or negative direction. If we modify the expression to \(y=\frac{1}{x}\), we would observe a horizontal asymptote along the line \(y=0\), as the value of the function gets closer and closer to zero but never reaches it, no matter how large \(x\) becomes.

Understanding asymptotes is crucial for graphing rational functions because it aids in predicting and illustrating the direction and behavior of the graph, especially near critical points where the function is undefined or at infinity.

Division by Zero

In arithmetic and algebra, division by zero is a situation where a number is divided by zero (0). It is undefined because the operation does not produce a unique or finite result, leading to a discontinuity in the function.

Let's break down the concept involved in the exercise. We see a rational function where the numerator \(y_{1}=x^{2}-2x\) and the denominator \(y_{2}=x\). Everything looks normal until we reach the point at which \(x=0\). At this point, we are attempting to compute \(y_{3}=\frac{y_{1}}{y_{2}}=\frac{x^{2}-2x}{0}\), which is an undefined operation.

This concept is essential in understanding why functions behave the way they do near certain values. When graphing, you will often see the function shoot off towards infinity, leading to what is known as a vertical asymptote, as the function value becomes larger and larger in either the positive or negative direction, depending on the numerator's sign.

Function Behavior Analysis

The analysis of a function's behavior involves looking deeply into how a function behaves near asymptotes, at intercepts, during intervals of increase and decrease, and at points of discontinuity. In rational functions, especially, understanding the relationship between numerators and denominators becomes critical.

In the exercise, we see that the behavior of \(y_{3}\) as it nears \(x=0\) is significant because it leads to a vertical asymptote. By using the TRACE feature as suggested in the steps, we can analyze this behavior: how the graph of the function fails to exist at the point of division by zero and hence, shoots off towards infinity.

Approaching from Positive and Negative Directions

When approaching an asymptote, it is important to consider both directions. As you approach \(x=0\) from the positive and negative sides, the values of \(y_{3}\) change dramatically. On one side, the values tend to positive infinity, and on the other, negative infinity, showcasing the different sides of the vertical asymptote.