Problem 21 - Thomas Calculus

Question

Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. \begin{equation} f(x)=\frac{x-1}{x^{2}-x-6} \end{equation}

Step-by-Step Solution

Answer

Set the window to \(-5 \leq x \leq 5\) and \(-10 \leq y \leq 10\).

Step 1: Factor the Denominator

First, we need to factor the quadratic expression in the denominator of the function \( f(x) = \frac{x-1}{x^2-x-6} \). The denominator \( x^2 - x - 6 \) can be factored as \( (x - 3)(x + 2) \).

Step 2: Identify Discontinuities

The function \( f(x) = \frac{x-1}{(x-3)(x+2)} \) is undefined where the denominator is zero. Therefore, the points \( x = 3 \) and \( x = -2 \) are vertical asymptotes. These are important discontinuities to consider when choosing the viewing window.

Step 3: Determine Horizontal Asymptote

As \( x \to \pm \infty \), the function behaves like \( \frac{x}{x^2} = \frac{1}{x} \), which approaches 0 as \( x \) goes to \( \pm \infty \). Thus, the line \( y = 0 \) is a horizontal asymptote.

Step 4: Consider the Behavior Near Asymptotes

Consider the behavior of the function near the asymptotes: around \( x = 3 \) and \( x = -2 \), the function dramatically increases or decreases, confirming the presence of vertical asymptotes.

Step 5: Set Graphing Software Viewing Window

A suitable window for viewing the overall behavior of this function can be \(-5 \leq x \leq 5\) for the x-axis and \(-10 \leq y \leq 10\) for the y-axis. This includes extreme changes near asymptotes and the horizontal asymptote along the axis.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines where a graph heads towards infinity or negative infinity. They are essentially boundaries that the function cannot touch or cross. To find these in a rational function, you look at the points that cause the denominator to be zero. For the function \( f(x) = \frac{x-1}{x^2-x-6} \), the denominator is \( x^2 - x - 6 \). Factoring this quadratic gives \((x-3)(x+2)\). The points where these factors equal zero, \(x = 3\) and \(x = -2\), are where the function is undefined. These become your vertical asymptotes.

• At \(x = 3\), the graph approaches but never touches this vertical line.
• At \(x = -2\), a similar behavior occurs.

Vertical asymptotes are crucial for understanding the function's behavior. They indicate where the function will "shoot" towards infinity or negative infinity, creating a gap or break in the graph.

Horizontal Asymptotes

Horizontal asymptotes help us understand the end behavior of a rational function, or how the function behaves as \(x\) goes to infinity or negative infinity. For the function \( f(x) = \frac{x-1}{x^2-x-6} \), analyzing the degrees of the polynomial in the numerator and the denominator helps find this line. Here, the numerator \(x-1\) has degree 1, and the denominator \(x^2-x-6\) has degree 2.

• When the degree of the denominator is higher than the numerator, the horizontal asymptote is \(y = 0\).

This tells us that as \(x\) goes to very large positive or negative numbers, the value of the function approaches zero. The line \(y = 0\) is the horizontal asymptote and indicates that the graph will get closer and closer to this line but doesn't necessarily ever touch it. It's like a guide showing the direction of the tails of the graph.

Factoring Quadratics

Factoring quadratics is a fundamental skill in algebra that involves expressing a quadratic expression as a product of its linear factors. This step is often crucial for finding the roots of the quadratic equation and can help simplify rational expressions. Take the quadratic \(x^2 - x - 6\). To factor this, we're looking for two numbers that multiply to \(-6\) (the constant term) and add up to \(-1\) (the coefficient of the middle term).

• \(x^2 - x - 6 = (x-3)(x+2)\), where \(-3\) and \(+2\) multiply to \(-6\) and add to \(-1\).

Factoring quadratics simplifies the process of finding vertical asymptotes for rational functions, as it aids in identifying where the denominator equals zero. This factorization is essential for both graphing and understanding discontinuities in rational expressions.