Problem 65 - Thomas Calculus

Question

Graph the rational functions in Exercises \(63-68\) . Include the graphs and equations of the asymptotes and dominant terms. $$y=\frac{1}{2 x+4}$$

Step-by-Step Solution

Answer

The graph of \(y=\frac{1}{2x+4}\) has a vertical asymptote at \(x=-2\). The graph is a hyperbola with no horizontal asymptote.

Step 1: Identify Asymptotes

To find the vertical asymptotes, we set the denominator equal to zero: \[ 2x + 4 = 0 \]Solving for \(x\), we find \(x = -2\). There are no horizontal asymptotes since the degree of the numerator is less than the degree of the denominator. However, we have an oblique asymptote for more complex cases, but in this case, it does not apply.

Step 2: Write Dominant Terms

Since the function \( y = \frac{1}{2x+4} \) does not have a polynomial of higher degree, the function itself serves as the dominant term. Therefore, we have:\[ y_{ ext{dominant}} = \frac{1}{2x+4} \].

Step 3: Graph the Function

Start by plotting the vertical asymptote at \(x = -2\). Then, plot several points around this asymptote to see the behavior of the function. For example, calculate \(y\) for \(x = 0\), \(x = -1\), \(x = -3\), etc. Draw the curve that approaches the vertical asymptote and behaves as expected at infinity.

Step 4: Verify and Summarize

Verify that the graph reflects the asymptote and makes sense with plotted points. Summarize that the asymptote equation is \(x = -2\) and that the graph is a hyperbola opening upwards and downwards, crossing the y-axis halfway at \( y = 0.25 \) when \( x = 0 \).

Key Concepts

Understanding Asymptotes

Asymptotes are critical elements of rational functions that help us comprehend their behavior as they reach toward infinity or a particular point. In essence, an asymptote is a line that the graph of a function approaches but never actually touches.

• There are mainly three types of asymptotes: vertical, horizontal, and oblique.
• Vertical asymptotes occur where the denominator of a rational function equals zero.
• Horizontal asymptotes describe the behavior of a function as it heads toward infinity in either the positive or negative direction.
• Oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator.

In our function, we only have a vertical asymptote, which forms a crucial aspect of its graph. Becasue it defines where the function moves infinitely upwards or downwards. This understanding aids us in sketching a more precise representation of the function.

Exploring Vertical Asymptotes

Vertical asymptotes occur at points where the denominator of a fraction in the function becomes zero, which makes the function undefined at those points. Identifying these spots helps predict where the function will surge towards positive or negative infinity.For the function given, \( y = \frac{1}{2x+4} \), determining the vertical asymptote is a straightforward process.

• First, set the denominator equal to zero: \( 2x + 4 = 0 \).
• Solving for \(x\) gives us \( x = -2 \).

This vertical line at \( x = -2 \) is a boundary where the function cannot exist, thus creating a split in the graph. On either side of this asymptote, the graph will tend rapidly towards infinity, shaping into a hyperbolic curve. Recognizing this feature lets us understand how the graph behaves and aids in predicting the movement near this asymptote.

Defining Dominant Terms

In the context of rational functions, dominant terms refer to the elements of the function that majorly influence its behavior, particularly as it approaches infinity. For the function \( y = \frac{1}{2x+4} \), identifying the dominant term is straightforward because the lack of a higher degree polynomial simplifies the equation:

• The entire function, \( y = \frac{1}{2x+4} \), acts as the dominant term.

In more complex functions, dominant terms are typically the parts with the highest degree in both the numerator and the denominator. These dictate how the function grows or shrinks, contributing to the horizontal or oblique asymptotes or lack thereof. For these types of functions, it is helpful to assess how the dominant terms interact, as they often determine the function’s end behavior, giving us better insights into its graph's long-term trends.