Problem 70 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{x^{2}-4}{x^{2}-x-6} $$

Step-by-Step Solution

Answer

Graph has a hole at \(x=-2\), a vertical asymptote at \(x=3\), and a horizontal asymptote at \(y=1\).

Step 1: Factor the Numerator and Denominator

The function given is \( f(x) = \frac{x^2 - 4}{x^2 - x - 6} \). Begin by factoring both the numerator and the denominator. The numerator \( x^2 - 4 \) is a difference of squares and can be factored as \((x - 2)(x + 2)\).The denominator \( x^2 - x - 6 \) factors into \((x - 3)(x + 2)\). So, the function becomes: \[ f(x) = \frac{(x - 2)(x + 2)}{(x - 3)(x + 2)} \].

Step 2: Simplify the Function and Identify Holes

Next, simplify the function by canceling the common factors in the numerator and denominator. The common factor \((x + 2)\) can be canceled out, leaving:\[ f(x) = \frac{x - 2}{x - 3} \]Since we canceled \((x + 2)\), there is a hole at \(x = -2\) in the graph of the function.

Step 3: Identify Vertical Asymptotes

To find vertical asymptotes, set the denominator equal to zero after simplification:\( x - 3 = 0 \)Solve for \(x\): \( x = 3 \)There is a vertical asymptote at \(x = 3\).

Step 4: Identify Horizontal Asymptotes

To find horizontal asymptotes, compare the degrees of the numerator and denominator. Both the numerator and denominator have a degree of 1 after simplification, so use the leading coefficients to determine the asymptote:The leading coefficient of the numerator is 1, and the denominator is also 1. Therefore, the horizontal asymptote is:\( y = \frac{1}{1} = 1 \).

Step 5: Graph the Function

Using the information obtained: - Plot a point where the hole occurs (\(x = -2\)) and denote the hole with an open circle on the graph.- Draw a vertical dashed line at \(x = 3\) for the vertical asymptote.- Draw a horizontal dashed line at \(y = 1\) for the horizontal asymptote.- Sketch the curve \(y = \frac{x-2}{x-3}\) around these asymptotes, ensuring it approaches the asymptotes without touching them.

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is an important skill in mathematics, particularly when working with rational functions. In the given function, the numerator is a quadratic expression: \[ x^2 - 4 \] This is a difference of squares and can be factored into: \[(x - 2)(x + 2)\]Similarly, the denominator \[ x^2 - x - 6 \]can be factored by finding two numbers that multiply to -6 and add to -1, which are -3 and 2. Thus, it factors to: \[(x - 3)(x + 2)\] Factoring allows us to simplify rational functions and is also essential for identifying asymptotes and holes in graphs. Quadratic expressions often appear in different forms, so always practice factoring them efficiently.

Vertical and Horizontal Asymptotes

Asymptotes are lines that the graph of a function can approach but never actually touch. There are two types that are important in rational functions: vertical and horizontal. Vertical asymptotes occur where the denominator of a simplified rational function equals zero, causing the function to be undefined at certain points. In our function \[ f(x) = \frac{(x - 2)(x + 2)}{(x - 3)(x + 2)} \]there's a vertical asymptote at \(x = 3\)since setting the simplified denominator \(x - 3 = 0\)leaves \(x = 3\)Horizontal asymptotes, on the other hand, are found by comparing the degrees of the numerator and denominator. In this problem, both the numerator \((x - 2)\) and the denominator \((x - 3)\) of the simplified function have a degree of one. Thus, the horizontal asymptote is determined by the ratio of their leading coefficients: \( y = \frac{1}{1} = 1 \).Horizontal asymptotes give an idea of how the function behaves as \(x\) approaches positive or negative infinity.

Graphing Rational Functions

Graphing rational functions involves understanding and applying the information obtained from factoring, identifying asymptotes, and analyzing holes. First, plot crucial components on the graph:

• Vertical asymptotes as dashed lines (e.g., \(x = 3\)).
• Horizontal asymptotes also as dashed lines (e.g., \(y = 1\)).

The graph of the function will approach these asymptotes, getting closer and closer but never touching them. For the function:\[ y = \frac{x-2}{x-3} \]sketch the curve such that it asymptotically approaches the vertical line where \(x = 3\) and the horizontal line where \(y = 1\). The function might exhibit different behavior on either side of the vertical asymptote, so ensure the curve reflects these asymptotic properties.

Identifying Holes in Graphs

Holes in the graph of a rational function occur when a factor is canceled out during simplification. In this specific exercise, there was a common factor \((x+2)\) in both the numerator and the denominator of \[ f(x) = \frac{(x-2)(x+2)}{(x-3)(x+2)} \]. Canceling this factor leaves a hole at \(x = -2\). It's crucial to remember that a hole is not an asymptote; it's a single, undefined point on the graph. To denote a hole on a graph, use an open circle at the \(x\)value of the hole. Recognizing holes can have significant impact on the behavior of the function's graph, so identifying them accurately ensures a complete understanding of the function's behavior.