Problem 54 - College Algebra Essentials

Question

Graph each equation. $$ y=-\frac{1}{x}\left(\text { Let } x=-2,-1,-\frac{1}{2},-\frac{1}{3}, \frac{1}{3}, \frac{1}{2}, 1, \text { and } 2 .\right) $$

Step-by-Step Solution

Answer

To graph \(y=-\frac{1}{x}\), create a table using the provided x-values, calculate their corresponding y-values, and plot these points on the graph.

Step 1: Create a Table of Values

Start by creating a table to represent the x-values and the corresponding y-values. The x-values have been provided (-2,-1,-1/2,-1/3,1/3,1/2,1,2). Set these up in a column in the table.

Step 2: Calculate the Y-Values

Now, calculate the y-values by substituting each x-value into the equation \(y=-\frac{1}{x}\). For example, for \(x=-2\), \(y=-\frac{1}{-2}=0.5\). Compute the rest of the y-values similarly.

Step 3: Plot the Values on the Graph

Once all x-values and y-values have been calculated, they can now be plotted on the graph. The graph of \(y=-\frac{1}{x}\) will be a hyperbola, and it should show the characteristic 'butterfly' shape when all points are connected.

Key Concepts

Hyperbola

When dealing with rational functions like the one in this exercise, it's helpful to understand the shape they take when graphed. The equation given, \( y = -\frac{1}{x} \), represents a type of curve called a hyperbola. In this case, the hyperbola has two separate branches that appear in opposite quadrants of the coordinate plane. This happens because the function is a rational one, with \( x \) in the denominator, which causes the graph to split into two distinct sections.

Hyperbolas are fascinating because they exhibit a specific symmetry. They have a vertical asymptote (a line that the graph approaches but never touches) and a horizontal asymptote. For \( y = -\frac{1}{x} \), the vertical asymptote is the \( y \)-axis (where \( x = 0 \)), and the horizontal asymptote is the \( x \)-axis (where \( y = 0 \)).

• Vertical Asymptote: This is where the graph can't be defined. As \( x \) gets very close to zero, \( y \) tends to infinity.
• Horizontal Asymptote: As \( x \) moves far to the left or right on the plane, \( y \) approaches zero.

Recognizing these asymptotes helps tremendously with sketching hyperbolas without plotting numerous points.

Table of Values

Before plotting a function on a graph, it's crucial to understand how to construct a table of values. This is a straightforward method to visualize how the equation behaves for different \( x \) values by calculating the corresponding \( y \) values. For the equation \( y = -\frac{1}{x} \), we can use the given x-values to complete this process.

For each \( x \) value, plug it into the equation to find \( y \). The sign and magnitude of \( y \) indicate the position of each point on the graph. Let's explore this with a few examples:

• For \( x = -2 \), \( y = -\frac{1}{-2} = 0.5 \).
• For \( x = 1 \), \( y = -\frac{1}{1} = -1 \).
• For \( x = 0.5 \), \( y = -\frac{1}{0.5} = -2 \).

Once you complete this for all x-values, you have a set of points that represent the curve on the coordinate plane. This makes graphing the curve much easier and provides a clear look at how the function behaves across different intervals.

Coordinate Plane Plotting

The process of plotting points on a coordinate plane is a fundamental skill in graphing rational functions. Once you've created your table of values and calculated the y-values, you're ready to transfer these points onto a graph.

To do this, start by plotting each point as an \((x, y)\) pair on the plane. Here’s how to make those calculations work:

• Locate each \( x \) value on the horizontal axis.
• Find the corresponding \( y \) value on the vertical axis.
• Mark the point where these two values intersect.

Connecting these points will reveal the hyperbola's distinct shape. Remember, the graph of \( y = -\frac{1}{x} \) doesn’t pass through the origin (0, 0) and shows symmetry as described above. Be mindful of the asymptotes: avoid drawing the curve on or along the lines of \( x = 0 \) and \( y = 0 \). This plotting method helps visualize the behavior and properties of rational functions effectively.