Problem 2 - 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 \(4 .\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 1.5 & {} \\ {1.9} & {} \\\ {1.99} & {} & {} \\ {1.999} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 2.5 & {} \\ {2.1} & {} & {} \\\ {2.01} & {} \\ {2.001} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 10 & {} \\ {50} & {} \\\ {100} & {} \\ {1000} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline-10 & {} \\ {-50} & {} \\ {-100} & {} \\ {-1000} & {} \\\ \hline\end{array}\) \(r(x)=\frac{4 x+1}{x-2}\)

Step-by-Step Solution

Answer

The vertical asymptote is at \(x = 2\); \(r(x)\) approaches \(-\infty\) from the left and \(+\infty\) from the right. The horizontal asymptote is \(y = 4\).

Step 1: Calculate Values Near the Vertical Asymptote (1.5 to 1.999)

The vertical asymptote occurs where the denominator equals zero, which is at \(x = 2\). Calculate \(r(x)\) for the given values approaching 2 from the left: \(x = 1.5\), \(1.9\), \(1.99\), and \(1.999\). For \(x = 1.5\),\[r(1.5) = \frac{4(1.5) + 1}{1.5 - 2} = \frac{7}{-0.5} = -14\]For \(x = 1.9\),\[r(1.9) = \frac{4(1.9) + 1}{1.9 - 2} = \frac{8.6 + 1}{-0.1} = -95\]For \(x = 1.99\),\[r(1.99) = \frac{4(1.99) + 1}{1.99 - 2} = \frac{8.96 + 1}{-0.01} = -996\]For \(x = 1.999\),\[r(1.999) = \frac{4(1.999) + 1}{1.999 - 2} = \frac{9.996 + 1}{-0.001} = -9996\]

Step 2: Calculate Values Near the Vertical Asymptote (2.5 to 2.001)

Now calculate the values as \(x\) approaches 2 from the right: \(x = 2.5\), \(2.1\), \(2.01\), and \(2.001\).For \(x = 2.5\),\[r(2.5) = \frac{4(2.5) + 1}{2.5 - 2} = \frac{10 + 1}{0.5} = 22\]For \(x = 2.1\),\[r(2.1) = \frac{4(2.1) + 1}{2.1 - 2} = \frac{8.4 + 1}{0.1} = 94\]For \(x = 2.01\),\[r(2.01) = \frac{4(2.01) + 1}{2.01 - 2} = \frac{8.04 + 1}{0.01} = 1004\]For \(x = 2.001\),\[r(2.001) = \frac{4(2.001) + 1}{2.001 - 2} = \frac{9.004 + 1}{0.001} = 10004\]

Step 3: Describe Behavior Near Vertical Asymptote

From the computed values, as \(x\) approaches 2 from the left (Table 1), \(r(x)\) decreases towards negative infinity. As \(x\) approaches 2 from the right (Table 2), \(r(x)\) increases towards positive infinity. Thus, the behavior at the vertical asymptote \(x = 2\) is that \(r(x)\) approaches \(-\infty\) from the left and \(+\infty\) from the right.

Step 4: Calculate Values for Horizontal Asymptote (Positive infinity)

Next, consider values as \(x\) approaches positive infinity in Tables 3, using \(x = 10\), \(50\), \(100\), and \(1000\).For \(x = 10\), \[r(10) = \frac{4(10) + 1}{10 - 2} = \frac{41}{8} = 5.125\]For \(x = 50\), \[r(50) = \frac{4(50) + 1}{50 - 2} = \frac{201}{48} = 4.1875\]For \(x = 100\),\[r(100) = \frac{4(100) + 1}{100 - 2} = \frac{401}{98} = 4.0918\]For \(x = 1000\),\[r(1000) = \frac{4(1000) + 1}{1000 - 2} = \frac{4001}{998} \approx 4.008\]

Step 5: Calculate Values for Horizontal Asymptote (Negative infinity)

Now consider values as \(x\) approaches negative infinity in Table 4, using \(x = -10\), \(-50\), \(-100\), and \(-1000\).For \(x = -10\), \[r(-10) = \frac{4(-10) + 1}{-10 - 2} = \frac{-39}{-12} = 3.25\]For \(x = -50\), \[r(-50) = \frac{4(-50) + 1}{-50 - 2} = \frac{-199}{-52} = 3.8269\]For \(x = -100\), \[r(-100) = \frac{4(-100) + 1}{-100 - 2} = \frac{-399}{-102} = 3.9118\]For \(x = -1000\), \[r(-1000) = \frac{4(-1000) + 1}{-1000 - 2} = \frac{-3999}{-1002} \approx 4\]

Step 6: Determine Horizontal Asymptote

The computed values from Tables 3 and 4 show that both as \(x\) approaches positive and negative infinity, \(r(x)\) approaches 4. Therefore, the horizontal asymptote of the function is \(y = 4\).

Key Concepts

Vertical Asymptote

A vertical asymptote is a line where a graph of a function becomes unbounded and heads towards infinity or negative infinity as it approaches the line. For rational functions, vertical asymptotes occur where the denominator of the function equals zero. In the given rational function \[ r(x) = \frac{4x + 1}{x - 2} \]we find the vertical asymptote by setting the denominator equal to zero: \[ x - 2 = 0 \]This gives us the vertical asymptote at \( x = 2 \). As \( x \) approaches 2 from the left (for example, 1.99 or 1.999), the function's value, \( r(x) \), decreases sharply towards negative infinity. As \( x \) approaches 2 from the right (such as 2.01 or 2.001), the function's value increases towards positive infinity. Therefore, at \( x = 2 \), the function has a vertical asymptote where it behaves very differently depending on whether you approach from the left or the right.

Horizontal Asymptote

Horizontal asymptotes refer to the value that the function approaches as \( x \) becomes very large in the positive or negative direction. In rational functions, the behavior of the function as \( x \) approaches infinity helps to identify the horizontal asymptote. For the function \[ r(x) = \frac{4x + 1}{x - 2} \]as \( x \) becomes very large, the terms with the highest degree, \( 4x \) in the numerator and \( x \) in the denominator, dominate the function. Simplifying this, the horizontal asymptote can be found by dividing the leading terms:\[ \lim_{{x \to \pm\infty}} r(x) = \lim_{{x \to \pm\infty}} \frac{4x}{x} = 4 \]Thus, the horizontal asymptote is \( y = 4 \). This tells us that no matter how large or small \( x \) becomes, \( r(x) \) will get closer and closer to 4, allowing us to understand the long term stability or destination of the function.

Behavior of Functions

Understanding the behavior of functions involves studying how functions act as \( x \) moves toward the vertical or horizontal asymptotes or infinity. Knowing this helps predict the path or trend of the graph of the function.### Near Vertical AsymptotesFor the function \[ r(x) = \frac{4x + 1}{x - 2} \]as \( x \) approaches 2:- From the left (\( x < 2 \)), the function behaves erratically and its value tends to negative infinity.- From right (\( x > 2 \)), it shoots up towards positive infinity.The graph would thus appear to diverge in opposite directions near \( x = 2 \).

### Near Horizontal AsymptotesFor large absolute values of \( x \), the function becomes more stable:- As \( x \) approaches either positive or negative infinity, \( r(x) \) approaches 4 steadily.

This asymptotic behavior helps us predict how the function behaves in extreme conditions, allowing for a better conceptual understanding of rational functions and their graph characteristics.