Problem 68 - College Algebra

Question

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote. $$ f(x)=\frac{2 x}{(x-3)^{2}} $$

Step-by-Step Solution

Answer

Vertical asymptote: \( x = 3 \); horizontal asymptote: \( y = 0 \).

Step 1: Identify the Asymptotes

The function \( f(x) = \frac{2x}{(x-3)^2} \) has a vertical asymptote where the denominator is zero, which is at \( x = 3 \). To find horizontal asymptotes, we analyze the behavior of the function as \( x \to \infty \) and \( x \to -\infty \). Since the degree of the polynomial in the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is \( y = 0 \).

Step 2: Create a Table for Vertical Asymptote \( x = 3 \)

Examine the behavior of \( f(x) \) as \( x \) approaches 3 from the left and right. Choose values getting closer to 3, such as 2.9, 2.99, 3.01, and 3.1, and calculate function values:\[\begin{array}{c|c}x & f(x) \\hline2.9 & 200 \2.99 & 20000 \3.01 & -20000 \3.1 & -200 \\end{array}\]The function increases without bound as \( x \to 3^- \), and decreases without bound as \( x \to 3^+ \).

Step 3: Create a Table Approaching the Horizontal Asymptote \( y = 0 \)

Calculate function values for large magnitudes of \( x \), like \( x = 10, 100, -10, -100 \):\[\begin{array}{c|c}x & f(x) \\hline10 & 0.025 \100 & 0.0002 \-10 & -0.025 \-100 & -0.0002 \\end{array}\]These indicate that \( f(x) \) approaches 0 as \( x \to \infty \) and \( x \to -\infty \).

Key Concepts

vertical asymptote

A vertical asymptote is a line that the graph of a function approaches but never touches as the function values become extremely large or small. It marks a point where the function stops being defined, usually because the denominator of a fraction becomes zero.
In the function \( f(x) = \frac{2x}{(x-3)^2} \), the vertical asymptote occurs when \( x = 3 \). Here, the denominator \((x-3)^2\) equals zero, making the function undefined at \( x = 3 \). The function behaves peculiarly near this point:

• As \( x \) approaches 3 from the left (\( x \to 3^- \)), \( f(x) \) increases without bound, shooting up to infinity.
• As \( x \) approaches 3 from the right (\( x \to 3^+ \)), \( f(x) \) decreases without bound, dropping down to negative infinity.

This is why, in our example, the function exhibits such dramatic changes near the vertical asymptote at \( x = 3 \).

horizontal asymptote

Horizontal asymptotes describe the behavior of a function as the input \( x \) becomes very large (\( x \to \infty \)) or very negative (\( x \to -\infty \)). They provide a boundary that the function values approach but do not necessarily reach.
For the function \( f(x) = \frac{2x}{(x-3)^2} \), the horizontal asymptote is at \( y = 0 \). This occurs because the degree of the polynomial in the numerator (1) is less than that in the denominator (2). As \( x \) gets larger in magnitude, the fraction \( \frac{2x}{(x-3)^2} \) tends to zero.

• This means as \( x \to \infty \), \( f(x) \to 0 \).
• Similarly, as \( x \to -\infty \), \( f(x) \to 0 \).

So, the graph of the function appears to flatten out to the horizontal line \( y = 0 \) as either \( x \to \infty \) or \( x \to -\infty \).

function behavior

Understanding the function's behavior near asymptotes is crucial in analyzing rational functions. The function \( f(x) = \frac{2x}{(x-3)^2} \) shows two types of asymptotic behavior:
The vertical asymptote at \( x = 3 \) divides the function into regions with different behaviors:

• To the left of the asymptote (\( x < 3 \)), the function exhibits extremely large positive values as \( x \to 3^- \).
• To the right of the asymptote (\( x > 3 \)), the values are large and negative, as \( x \to 3^+ \).

On the other hand, horizontal asymptotes indicate that no matter how large or small the \( x \)-values are, \( f(x) \) will come close to but not diverge from a certain value:

• As \( x \to \infty \) or \( x \to -\infty \), the function's output nears 0, which is consistent with the horizontal asymptote \( y = 0 \).

This dual asymptotic behavior forms a clear picture of how \( f(x) \) behaves across its entire range.

rational functions

Rational functions are quotients of polynomials of the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. They can model many real-world phenomena and have unique features like asymptotes.
For instance, the rational function \( f(x) = \frac{2x}{(x-3)^2} \) has specific characteristics due to its form:

• **Vertical Asymptotes**: Where \( Q(x) = 0 \). Here, the asymptote \( x = 3 \) occurs because \( (x-3)^2 \) becomes zero when \( x \) is 3.
• **Horizontal Asymptotes**: Determined by comparing the degrees of \( P(x) \) and \( Q(x) \). Since the degree of \( P(x) \) is less than \( Q(x) \), the horizontal asymptote is \( y = 0 \).

These functions often exhibit complex behaviors near their asymptotes, requiring close examination of both numerator and denominator polynomials to predict these patterns.