Problem 3 - Algebra and Trigonometry

Question

A rational function is given. (a) Complete each table for the function. (b) Describe the behavior of the function nea its vertical asymptote, based on Tables 1 and \(2 .\) (c) Determine the horizontal asymptote, based on Tables 3 and \(4 .\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 1.5 & {} \\ {1.9} & {} \\\ {1.99} & {} & {} \\ {1.999} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 2.5 & {} \\ {2.1} & {} & {} \\\ {2.01} & {} \\ {2.001} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 10 & {} \\ {50} & {} \\\ {100} & {} \\ {1000} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline-10 & {} \\ {-50} & {} \\ {-100} & {} \\ {-1000} & {} \\\ \hline\end{array}\) \(r(x)=\frac{3 x-10}{(x-2)^{2}}\)

Step-by-Step Solution

Answer

Vertical asymptote at \( x = 2 \); horizontal asymptote at \( y = 0 \).

Step 1: Evaluate Function for Table 1

The function given is \( r(x) = \frac{3x-10}{(x-2)^2} \). To fill in Table 1, we will plug in the values of \( x \) into the function. For \( x = 1.5 \), \( r(1.5) = \frac{3(1.5)-10}{(1.5-2)^2} = \frac{4.5 - 10}{0.25} = \frac{-5.5}{0.25} = -22 \). Similarly, calculate for \( x = 1.9, 1.99, 1.999 \).

Step 2: Evaluate Function for Table 2

For Table 2, use values \( x = 2.5, 2.1, 2.01, 2.001 \). For \( x = 2.5 \), \( r(2.5) = \frac{3(2.5)-10}{(2.5-2)^2} = \frac{7.5 - 10}{0.25} = \frac{-2.5}{0.25} = -10 \). Continue the calculation for the remaining values of \( x \) in the table.

Step 3: Describe Behavior Near Vertical Asymptote

The vertical asymptote occurs where the denominator is zero, i.e., at \( x = 2 \). By examining Tables 1 and 2, as \( x \to 2^- \) or \( x \to 2^+ \), \( r(x) \) goes to \( -\infty \) or \( +\infty \), indicating a vertical asymptote at \( x = 2 \).

Step 4: Evaluate Function for Table 3

For Table 3, we calculate \( r(x) \) for large positive \( x = 10, 50, 100, 1000 \). For \( x = 10 \), \( r(10) = \frac{3(10)-10}{(10-2)^2} = \frac{30 - 10}{64} = \frac{20}{64} = 0.3125 \). Repeat this for the other values, ensuring we approach positive infinity.

Step 5: Evaluate Function for Table 4

For Table 4, use large negative \( x = -10, -50, -100, -1000 \). For \( x = -10 \), \( r(-10) = \frac{3(-10)-10}{(-10-2)^2} = \frac{-30 - 10}{144} = \frac{-40}{144}\approx -0.278 \). Compute similarly for the rest, noting approach to negative infinity.

Step 6: Determine Horizontal Asymptote

The horizontal asymptote can be inferred from Tables 3 and 4. As \( x \to +\infty \) or \( x \to -\infty \), \( r(x) \) approaches 0, indicating the horizontal asymptote is \( y = 0 \).

Key Concepts

Vertical Asymptote

In the realm of rational functions, understanding vertical asymptotes is crucial. Vertical asymptotes occur when the denominator of a rational function approaches zero, causing the function itself to "blow up," or head towards infinity. For the given function, \( r(x) = \frac{3x-10}{(x-2)^2} \), the denominator becomes zero at \( x = 2 \). This is where you will encounter a vertical asymptote.

To grasp the behavior around a vertical asymptote, you can observe the values of \( x \) nearing \( 2 \) from both sides.

• When \( x \) approaches 2 from the left (\( x \to 2^- \)), the function \( r(x) \) decreases towards \(-\infty\).
• Conversely, when \( x \) comes close from the right (\( x \to 2^+ \)), the function \( r(x) \) increases towards \(+\infty\).

By knowing this, you can anticipate significant changes and sudden spikes in graphs of the function at these points.

Horizontal Asymptote

Horizontal asymptotes are another fascinating feature of rational functions. They provide a sense of the function's behavior as \( x \) either ascends or descends to infinity. A horizontal asymptote can be interpreted as the y-value where the function is headed when \( x \) heads towards extreme values.

For our given function, analyzing the results from large \( x \) values in Tables 3 and 4 reveals that as \( x \to +\infty \) or \( x \to -\infty \), \( r(x) \) tends towards zero. This implies the presence of a horizontal asymptote at \( y = 0 \).

This means that far away from the origin, the function behaves like the line \( y = 0 \), getting increasingly flat as it stretches out along the x-axis. Horizontal asymptotes illustrate the end behavior, giving insights into how a function levels off over wide spans.

Function Behavior

Function behavior refers to how a function acts as \( x \) approaches certain values, either within or beyond the scope of immediate observation. In equations involving rational functions, behavior analysis especially makes a difference near asymptotes.

For the function \( r(x) = \frac{3x-10}{(x-2)^2} \), understanding how it behaves as \( x \) comes near the vertical asymptote, \( x = 2 \), or as \( x \) heads towards vast positive or negative numbers is crucial.

• Near the vertical asymptote: the function heads into extremes — enormous positive or negative values.
• Far from the origin, it calms and heads towards the horizontal asymptote of \( y = 0 \).

This duality of skyrocketing near critical points and stabilizing at extremes is fundamental in analyzing rational functions.

Asymptotic Analysis

Asymptotic analysis goes beyond establishing the mere presence of asymptotes. It dives into determining how the rational function approaches these asymptotic boundaries. For \( r(x) = \frac{3x-10}{(x-2)^2} \), asymptotic analysis involves quantifying how fast or slow the function moves towards its asymptotes.

By examining values of \( x \) near the vertical asymptote \( x = 2 \), you notice rapid escalation in function values as they venture close to infinity. Meanwhile, considering large positive or negative \( x \) values beyond these critical points, the function gently grazes the horizontal asymptote of \( y = 0 \).

This kind of analysis is vital in understanding the scaling behavior of functions well beyond local extremities, granting predictive power over their behavior in extremes.