Problem 6 - A Graphical Approach to College Algebra

Question

Provide a short answer to each question. Do not use a calculator. What is the equation of the vertical asymptote of the graph of \(y=\frac{1}{(x+2)^{2}}-4 ? \quad\) of the horizontal asymptote?

Step-by-Step Solution

Answer

Vertical asymptote: \( x = -2 \); Horizontal asymptote: \( y = -4 \).

Step 1: Identify the vertical asymptote

A vertical asymptote exists where the function is undefined. For the given function \( y = \frac{1}{(x+2)^2} - 4 \), the denominator \( (x+2)^2 \) cannot be zero. Set \( (x+2)^2 = 0 \) to find the undefined point. Solving this, we get \( x + 2 = 0 \), therefore \( x = -2 \). Thus, the equation of the vertical asymptote is \( x = -2 \).

Step 2: Determine the horizontal asymptote

The horizontal asymptote of a rational function is determined by the behavior of the function as \( x \to \infty \) or \( x \to -fty \). Observe that as \( x \) becomes very large, the term \( \frac{1}{(x+2)^2} \) approaches zero, because the denominator grows much faster than the numerator. Therefore, the function approaches \( y = 0 - 4 \), which simplifies to \( y = -4 \). Thus, the horizontal asymptote is \( y = -4 \).

Key Concepts

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. They provide essential insights into the behavior of a function as we delve into very large or small values of its variables.

There are two main types of asymptotes we generally discuss in algebra: **vertical asymptotes** and **horizontal asymptotes**.

• **Vertical Asymptotes** occur where the function is undefined. They are associated with the values of the variable that make the denominator zero. This can cause the function to grow indefinitely large (positive or negative). In our function, \(y = \frac{1}{(x+2)^2} - 4\), we set \((x+2)^2 = 0\). Solving gives \(x = -2\). Thus, the vertical asymptote is \(x = -2\).
• **Horizontal Asymptotes** illustrate the behavior of a function as the independent variable approaches infinity or negative infinity. In our function, as \(x\) becomes very large, \(\frac{1}{(x+2)^2}\) diminishes towards zero, letting \(y\) approach \(-4\). This gives a horizontal asymptote at \(y = -4\).

Rational Functions

Rational functions are expressed as the quotient of two polynomials. They have distinct properties that make them interesting to study, particularly in graphing.

For our provided example, \(y = \frac{1}{(x+2)^2} - 4\), it is highlighted that the function takes the form of a rational expression due to its polynomial structures in the numerator and denominator.

• The numerator is \(1\), which is a constant polynomial. In rational functions, a constant like this means the function's behavior is highly influenced by the denominator.
• The denominator is \((x+2)^2\), influencing both the location of vertical asymptotes and the function's overall behavior.

In general, rational functions involve interesting behaviors like asymptotes and sometimes undefined points, which students explore in problems like this one.

Graph Analysis

Graph analysis involves understanding how the graph of a function behaves across the entire set of possible values for its variables.

When analyzing the graph of our function, \(y = \frac{1}{(x+2)^2} - 4\), several key features emerge:

• **Shape and Behavior**: The \(\frac{1}{(x+2)^2}\) term ensures the function is never negative because squares are non-negative. Subtracting 4 shifts the entire graph downward by 4 units.
• **Asymptotes and Limits**: The previously discussed asymptotes appear on the graph as guidelines, constraining the trajectory of the curve without it ever crossing these lines.
• **Intercepts**: By setting \(y\) to zero, students can find the exact points where the graph crosses the \(x\)-axis, though in this question finding the zeros wasn’t the primary focus.

The graphical representation assists in visualizing these interactions, helping students develop robust intuition for how changes in the formula impact the graph's shape and limits.