Problem 13 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. $$ F(x)=\frac{1}{x} $$

Step-by-Step Solution

Answer

Plot the points and sketch a hyperbola with a vertical asymptote at \( x = 0 \).

Step 1: Choose x-values for the table

We start by selecting a range of x-values to substitute into the function. Since \( F(x) = \frac{1}{x} \) is undefined at \( x = 0 \), we'll choose values around zero. Select a mix of positive and negative values such as: \(-2, -1, -0.5, -0.1, 0.1, 0.5, 1, 2\).

Step 2: Calculate F(x) for each x-value

Substitute each selected x-value into the function \( F(x) = \frac{1}{x} \) to find corresponding y-values:- \( F(-2) = \frac{1}{-2} = -0.5 \)- \( F(-1) = \frac{1}{-1} = -1 \)- \( F(-0.5) = \frac{1}{-0.5} = -2 \)- \( F(-0.1) = \frac{1}{-0.1} = -10 \)- \( F(0.1) = \frac{1}{0.1} = 10 \)- \( F(0.5) = \frac{1}{0.5} = 2 \)- \( F(1) = \frac{1}{1} = 1 \)- \( F(2) = \frac{1}{2} = 0.5 \)

Step 3: Create a table of values

Organize the x-values and their corresponding F(x) values into a table: | x | F(x) | |-----|------| | -2 | -0.5 | | -1 | -1 | | -0.5| -2 | | -0.1| -10 | | 0.1 | 10 | | 0.5 | 2 | | 1 | 1 | | 2 | 0.5 |

Step 4: Sketch the graph

Plot each point from the table on a coordinate plane. Observe the behavior as x approaches 0 from both sides: \( F(x) \) moves towards \( -\infty \) as \( x \) approaches 0 from the left, and towards \( \infty \) as \( x \) approaches 0 from the right. Draw a smooth curve through the points, with the understanding there is a vertical asymptote at \( x = 0 \).

Key Concepts

Vertical Asymptote

In graphing rational functions, the concept of a vertical asymptote is crucial. Vertical asymptotes occur when the function approaches infinity (positive or negative) and are typically represented by a vertical line on the graph.

For the function \( F(x) = \frac{1}{x} \), the vertical asymptote is at \( x = 0 \). This means that the function is undefined at \( x = 0 \) because division by zero is not possible. As \( x \) gets closer to 0 from either the positive or negative side, the value of \( F(x) \) increases rapidly towards infinity or decreases towards negative infinity, which cannot be shown by a finite point on the graph.

When sketching the graph, it is important to draw a dotted or dashed vertical line at \( x = 0 \) to indicate the vertical asymptote. This will guide you in understanding how the function behaves as \( x \) approaches this critical point.

Table of Values

Creating a table of values is a helpful tool for sketching the graph of a function. This involves selecting various \( x \)-values and calculating the corresponding \( F(x) \)-values.

For the function \( F(x) = \frac{1}{x} \), a table of values allows us to observe how the function behaves at different points around the vertical asymptote, \( x = 0 \).

• Choose a range of \( x \)-values that include both positive and negative numbers, and values close to the asymptote.
• Calculate the corresponding \( y \)-values or \( F(x) \) values for each chosen \( x \)-value.
• Record these pairs in a table to visualize the data easily.

This table becomes a roadmap for plotting points on the graph and helps in drawing an accurate representation of the function.

Coordinate Plane

Once you have your table of values, the next step is to plot these points on a coordinate plane. The coordinate plane consists of two intersecting lines, the x-axis (horizontal) and the y-axis (vertical), which help locate points based on their \( x \) and \( y \) values.

For the function \( F(x) = \frac{1}{x} \), you'll place each corresponding pair from the table: \((x, F(x)) \) on the plane.

• Locate each \( x \)-value on the x-axis and \( F(x) \)-value on the y-axis.
• Plot a point where the two lines meet.

Continue this process for all points in the table. This visual guide helps you see the shape of the graph, and together with the vertical asymptote line, explains the function's overall behavior.

Function Behavior Near Asymptotes

Understanding function behavior near asymptotes is essential. As we saw, the function \( F(x) = \frac{1}{x} \) has a vertical asymptote at \( x = 0 \). The fascinating part about asymptotes is how they show the function's tendency as \( x \) gets infinitely close to the asymptote.

When \( x \) approaches 0 from the negative side (left), \( F(x) \) becomes significantly negative, moving rapidly towards \(-\infty\). Conversely, when \( x \) approaches 0 from the positive side (right), \( F(x) \) turns positive, increasing steeply towards \(\infty\). This drastic change in behavior is why asymptotes are important: they capture the idea that while the function can't quite reach these extreme values, it gets very close. This behavior is signified by the curve getting closer and closer to the asymptote without touching it.

Visualizing this behavior through graphing assists in deepening the understanding of how rational functions operate around asymptotes.