Problem 53 - Precalculus 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

Graph the given values on a coordinate plane to create a hyperbola that approaches but does not touch the x and y axes.

Step 1: Understanding the equation

The given equation is \( y = \frac{1}{x} \). This equation describes an hyperbola. The behavior of the hyperbola is such that it approaches, but never reaches, the x-axis or the y-axis.

Step 2: Calculate y for each x-value

Now you will substitute each x-value into the equation to obtain the corresponding y-value. The pairs (x, y) will then coorespond to the points that are to be plotted on the graph

Step 3: Create a table

List all x-values on one column, calculate the corresponding y-values and list them in another column. Your table should look something like this: \[ \begin{array}{|c|c|} \hline x & y \ \hline -2 & -0.5 \ -1 & -1 \ -\frac{1}{2} & -2 \ -\frac{1}{3} & -3 \ \frac{1}{3} & 3 \ \frac{1}{2} & 2 \ 1 & 1 \ 2 & 0.5 \ \hline \end{array} \]

Step 4: Plotting the Graph

Draw an x-y coordinate system. For each pair of (x, y) from your table, plot a point on the graph. Connect the points to create the hyperbola, making sure it approaches but never touches the axes.