Problem 67 - College Algebra

Question

For the following exercises, make tables to behavior of the function near the vertical asymptote and reffecting the horizontal asymptote $$ f(x)=\frac{2 x}{x+4} $$

Step-by-Step Solution

Answer

Vertical asymptote is at x = -4; horizontal asymptote at y = 2. Reflection is at y = -2.

Step 1: Determine the Vertical Asymptote

To find the vertical asymptotes of the function, set the denominator equal to zero and solve for \(x\). For \(f(x) = \frac{2x}{x+4}\), we have the equation \(x + 4 = 0\). Thus, \(x = -4\) is the vertical asymptote.

Step 2: Investigate Behavior Near the Vertical Asymptote

To understand the behavior of the function near \(x = -4\), consider values of \(x\) approaching \(-4\) from both the left \(x \to -4^-\) and the right \(x \to -4^+\). Construct a table with \(x\) values like \(-4.1\), \(-4.01\) for \(x \to -4^-\) and \(-3.9\), \(-3.99\) for \(x \to -4^+\). Compute \(f(x)\) for these values and note whether \(f(x)\) approaches \(\pm \infty\).

Step 3: Find the Horizontal Asymptote

For horizontal asymptotes, examine the behavior as \(x \to \pm \infty\). Divide the coefficients of the highest power terms of the function. Here, \(\lim_{{x \to \infty}} \frac{2x}{x+4} = \lim_{{x \to \infty}} \frac{2}{1} = 2\). So, the horizontal asymptote is \(y = 2\).

Step 4: Reflecting the Horizontal Asymptote

When a function's horizontal asymptote is given by \(y = a\), its reflection would be \(y = -a\). Here, with \(y = 2\) as the original horizontal asymptote, its reflection is \(y = -2\). This reflection does not change the vertical asymptote but gives insight into symmetry or inverse characteristics of a function.

Step 5: Graphic Representation of Asymptotes

Visually check and include the asymptotes while graphing the function. Draw vertical dashed lines on the graph at \(x = -4\) for the vertical asymptote and horizontal dashed lines at \(y = 2\) and \(y = -2\) (if considering the reflection) for the horizontal asymptotes.Observe how \(f(x)\) behaves approaching these lines without touching them.

Key Concepts

Vertical Asymptote

A vertical asymptote is a vertical line on a graph where a function's value increases or decreases without bound as the input approaches a specific value. For the given function, \( f(x) = \frac{2x}{x+4} \), you can find the vertical asymptote by setting the denominator equal to zero and solving for \( x \).

• Equation: \( x + 4 = 0 \)
• Solution: \( x = -4 \)

Thus, \( x = -4 \) is the vertical asymptote.To really understand this behavior, think of \( x \) getting closer to \(-4\) from both sides. For \( x \to -4^- \), or from the left, and \( x \to -4^+ \), or from the right, observe how \( f(x) \) behaves. As you plug in numbers approaching \(-4\) like \(-4.1\) or \(-3.9\), you'll see \( f(x) \) trends towards infinity or negative infinity. This pattern shows the function cannot actually reach \( x = -4 \), and highlights why it's useful to recognize vertical asymptotes on a graph.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that illustrates the behavior of a graph as the inputs (\( x \)) become very large or very small. In our function \( f(x) = \frac{2x}{x+4} \), to find the horizontal asymptote, examine the behavior of the function as \( x \to \pm \infty \).To do this, compare the degrees of the numerator and the denominator:

• The numerator is \( 2x \), and the denominator is \( x + 4 \), both of degree 1.

Their degrees are equal, so divide the leading coefficients, which are 2 in the numerator and 1 in the denominator:\[ \lim_{{x \to \infty}} \frac{2x}{x+4} = \frac{2}{1} = 2 \]Therefore, the horizontal asymptote is \( y = 2 \). This asymptote indicates that as \( x \) gets very large or very small, the function approaches this line but never quite reaches it directly, much like how an airplane coming in to land approaches the ground steadily without plummeting.

Limits in Calculus

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a specific point. This is especially useful when exploring asymptotes, as they heavily rely on the idea of approaching without precisely reaching.In the context of vertical asymptotes, limits help us analyze how a function behaves as it nears a specific point. For the function \( f(x) = \frac{2x}{x+4} \), you might be asked to find \( \lim_{{x \to -4^-}} f(x) \) and \( \lim_{{x \to -4^+}} f(x) \), illustrating how the function behaves near its vertical asymptote.For horizontal asymptotes, limits allow us to see what happens to \( f(x) \) as \( x \to \pm \infty \). In our example,\[ \lim_{{x \to \infty}} f(x) = 2 \]This limit shows that, as \( x \) increases or decreases indefinitely, the function value approaches 2. Limits give us a powerful tool to articulate and predict the behavior of functions, and they become essential when understanding both vertical and horizontal asymptotes.