Problem 53 - College Algebra

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

By inserting the specified values into the equation and finding their corresponding y-coordinates, seven points (seven because \(x=0\) is undefined in the equation) are marked on the graph. The curve connecting these points forms the graph of \(y=\frac{1}{x}\). The graph indicates that as x approaches 0 from either direction, y approaches +- infinity, creating a vertical asymptote at x=0. And as x approaches +- infinity, y approaches 0, creating a horizontal asymptote at y=0.

Step 1 - Calculation for negative values

First, insert the negative values from the set (-2, -1, -1/2, -1/3) one by one into the equation. To do this, replace x in \(y=\frac{1}{x}\) with each of these values and calculate.

Step 2 - Plot negative values

After calculating, the respective pairs of x and y coordinates can be plotted on the graph. Find the appropriate positions on the negative part of the x-axis and mark the coordinates.

Step 3 - Calculation for positive values

Following the same process as for negative values, insert the positive values (1/3, 1/2, 1, 2) into the equation \(y=\frac{1}{x}\). Calculate to get the corresponding y-values.

Step 4 - Plot the positive values

After acquiring the y-coordinates for positive x-values, they too can be plotted on the graph. Find the appropriate positions on the positive part of the X-axis f and mark the coordinates.

Step 5 - Graph the Equation

With all points marked, they can be connected to form the final graph of the equation. Since \(y=\frac{1}{x}\) is continuous, draw a smooth curve.