Problem 37 - Thomas Calculus

Question

Another way to avoid incorrect connections when using a graphing device is through the use of a “dot mode,” which plots only the points. If your graphing utility allows that mode, use it to plot the functions in Exercises 37–40. $$ y=\frac{1}{x-3} $$

Step-by-Step Solution

Answer

Use dot mode on the graphing device for \( y = \frac{1}{x-3} \) to plot points, avoiding connections that ignore the vertical asymptote at \( x = 3 \).

Step 1: Identify the Function and Limitations

We are given the function \( y = \frac{1}{x-3} \). Notice that this function has a vertical asymptote at \( x = 3 \) because division by zero is undefined.

Step 2: Set up Dot Mode on Graphing Device

To avoid incorrect connections that occur in continuous graphing modes, activate the 'dot mode' on your graphing utility. This mode will plot only discrete points of the function without connecting them, making it ideal for visualizing functions with discontinuities.

Step 3: Choose and Plot Points

Select a range of \( x \) values close to the vertical asymptote but avoiding it, such as \( x = 2, 2.5, 4, 5 \). Calculate \( y \) for each: \( y(2) = -1 \), \( y(2.5) = -2 \), \( y(4) = 1 \), and \( y(5) = 0.5 \). Plot these points using your graphing device in dot mode.

Step 4: Analyze the Graph

Once the points are plotted, observe how the function behaves. Note the steep increase or decrease of \( y \) values as \( x \) approaches 3 from both left and right, confirming the presence of a vertical asymptote at \( x = 3 \).

Key Concepts

Vertical Asymptotes

In the context of graphing rational functions, a vertical asymptote is a line that the graph approaches but never touches or crosses. This occurs in rational functions like \[ y = \frac{1}{x-3} \] where the denominator is zero at certain points, making the function undefined. For this function, the vertical asymptote occurs at \( x = 3 \). Why? Because substituting 3 into the denominator gives zero, which leads to division by zero—an undefined mathematical operation. Understanding vertical asymptotes is crucial as they indicate a boundary or limit in the function's graph, where the values can increase or decrease indefinitely as they near the line.

When sketching the graph manually, remember:

• Identify where the denominator equals zero.
• These 'zero points' indicate vertical asymptotes.
• As \( x \) approaches these numbers, \( y \) tends to infinity or minus infinity.

This understanding helps in graphing the behavior of the function near the asymptote without mistakenly connecting the points across this non-existing line.

Dot Mode in Graphing Calculators

Graphing calculators offer various modes to visualize functions effectively. One powerful feature is the "dot mode." This mode is essential when plotting functions with vertical asymptotes or other discontinuities as it avoids drawing misleading connections between plotted points. In our function \[ y = \frac{1}{x-3} \] dot mode will plot only the selected values, emphasizing the function's behavior without attempting to 'connect the dots' improperly.

Setting up dot mode can usually be done through the settings menu of your graphing calculator. Once activated, select a few points around the vertical asymptote (but not the asymptote itself) to visualize the gaps where the function is not defined. This helps:

• Avoid false representation of the graph’s continuity.
• Better understanding of the effect of the asymptote on the graph.
• A focused visualization on where the graph shoots up or down as it approaches the asymptote.

Using dot mode is a valuable technique for students trying to understand complex graph relationships particularly in rational functions.

Discontinuities in Graphs

When examining functions like \[ y = \frac{1}{x-3} \] discontinuities become apparent through vertical asymptotes, where the graph 'breaks' and doesn't connect smoothly. Discontinuities in graphs indicate where the function does not have a well-defined value, often leading to sharp jumps or vertical areas that the graph simply cannot cross.

Three main kinds of discontinuities can be encountered:

• **Removable Discontinuity**: When a hole exists in the graph, often due to a factor in numerator and denominator cancelling out.
• **Jump Discontinuity**: Seen in piecewise functions where the function jumps from one value to another without connecting lines.
• **Infinite Discontinuity**: Often exists at vertical asymptotes like in our function where the values shoot infinitely into positive or negative infinity.

Discontinuities tell us much about a function's domain and range and how that function behaves as \( x \) navigates through all real numbers excluding the points of discontinuity. Recognizing these will enhance the accuracy in visual interpretations and mathematical solutions.