Question

Complete the table for the function \(y=\frac{x+4}{x^2-16}\) Then use the trace feature of a graphing calculator to explain the behavior of the function at x = ?4. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline-3.5 & \\ \hline-3.8 & \\ \hline-3.9 & \\ \hline-4.1 & \\ \hline-4.2 & \\ \hline \end{array} $$

Short answer

The y-values corresponding to the given x-values in the function \(y=\frac{x+4}{x^2-16}\) are as follows: for x=-3.5, y=-0.20618557; for x=-3.8, y=-0.36443149; for x=-3.9, y=-0.89686099; for x=-4.1, y=0.81234768; and for x=-4.2, y=0.33259912. When \(x=-4\), the function is undefined as it results in a division by zero. The function has a vertical asymptote at \(x=-4\).

Step-by-step solution

Evaluate the function at the given 'x' values

The 'x' values to be evaluated in the function are -3.5, -3.8, -3.9, -4.1, -4.2. Calculate the value of 'y' for each 'x' value by substituting the x-value in the function equation \(y=\frac{x+4}{x^2-16}\). Below are the calculated y values corresponding to each x value.\n-3.5: \(y=\frac{-3.5 + 4}{-3.5^2 - 16} = -0.20618557\)\n-3.8: \(y=\frac{-3.8 + 4}{-3.8^2 - 16} = -0.36443149\)\n-3.9: \(y=\frac{-3.9 + 4}{-3.9^2 - 16} = -0.89686099\)\n-4.1: \(y=\frac{-4.1 + 4}{-4.1^2 - 16} = 0.81234768\)\n-4.2: \(y=\frac{-4.2 + 4}{-4.2^2 - 16} = 0.33259912\)

Analyze the behavior of the function at x = -4

If we look at the function's formula, when \(x=-4\), the denominator of the fraction becomes 0, since \((-4)^2-16=0\). This would result in a division by zero, which is undefined. Hence, the function is undefined at \(x=-4\). This will show up as a vertical asymptote (a vertical line where the function doesn't exist) on the graph at \(x=-4\). We can observe this from the surrounding points as well; as the value of 'x' approaches -4 from both sides, the value of 'y' increases or decreases without bound.

Complete the table

Now we can fill out the table with the evaluated y-values:\n $$ \n\begin{{array}}{{|c|c|}} \n\hline \boldsymbol{{x}} & \boldsymbol{{y}} \\ \n\hline-3.5 & -0.20618557\\ \n\hline-3.8 & -0.36443149 \\ \n\hline-3.9 & -0.89686099 \\ \n\hline-4.1 & 0.81234768 \\ \n\hline-4.2 & 0.33259912 \\ \n\hline \n\end{{array}} \n$$

Key concepts

Vertical Asymptote

In rational functions, a vertical asymptote is a line that the graph of the function approaches but never touches. In this exercise, we are dealing with the function \( y=\frac{x+4}{x^2-16} \). To find when a vertical asymptote occurs, you need to identify when the denominator equals zero. This is because division by zero is undefined in mathematics, causing the function's output to go to infinity or negative infinity.

In our case, the denominator \( x^2-16 \) equals zero when \( x=\pm 4 \). Therefore, the function has vertical asymptotes at \( x=4 \) and \( x=-4 \). This means that as \( x \) moves closer to \(-4\) from either direction, the \( y \)-value of the function will increase or decrease without any bound. This behavior reflects on the graph, appearing as sharp spikes shooting up or down along the vertical lines \( x=4 \) and \( x=-4 \).

Undefined Values

Undefined values in a function occur when the function cannot produce a result; this usually happens when there is division by zero. For the given function \( y=\frac{x+4}{x^2-16} \), undefined values occur when the denominator \( x^2-16 \) equals zero.

Solving \( x^2-16=0 \) gives us \( x^2=16 \). Hence, \( x=4 \) or \( x=-4 \). At these points, the function becomes undefined because it involves division by zero, resulting in values that are not valid in real-number terms. It's essential to recognize these points when analyzing functions, as they are locations where the graph may have holes or asymptotic behavior.

These undefined values are crucial for understanding where the function doesn't exist and how it behaves around those points.

Graphing Calculator

A graphing calculator is a powerful tool commonly used to visualise mathematical functions, equations, and graphs dynamically. It helps in understanding complex functions like rational expressions by easily showing asymptotes, holes, and undefined regions.

In this exercise, the graphing calculator can be used to input the function \( y=\frac{x+4}{x^2-16} \) and trace its behavior as \( x \) approaches \(-4\). By using the trace function, you can observe how the \( y \)-values of the function rise or fall dramatically as \( x \) nears \(-4\). This graphical insight affirms what's theoretically expected from vertical asymptotes and undefined values.

By visually exploring the graph, it becomes easier to comprehend the concept of asymptotes and where the function does not exist. The graphing calculator thereby solidifies understanding through an interactive graphical representation.

Function Evaluation

Function evaluation involves calculating the value of a function for specific inputs. It is a core technique used to understand the nature and behavior of mathematical expressions. For this exercise, we evaluate the function \( y=\frac{x+4}{x^2-16} \) at specific x-values, such as \(-3.5, -3.8, -3.9, -4.1,\) and \(-4.2\).

To perform this evaluation, substitute each \( x \)-value into the function. For instance, substituting \( x = -3.5 \) gives \( y = \frac{-3.5 + 4}{-3.5^2 - 16} \approx -0.20618557 \). This process is repeated for each given \( x \)-value to understand how the function behaves in those regions

• For \( x = -3.5 \), \( y \approx -0.206 \)
• For \( x = -3.8 \), \( y \approx -0.364 \)
• For \( x = -3.9 \), \( y \approx -0.897 \)
• For \( x = -4.1 \), \( y \approx 0.812 \)
• For \( x = -4.2 \), \( y \approx 0.333 \)

These calculations help map out parts of the graph, note changes in direction, and establish behavioral patterns around the asymptotes.