Question

Graph$y=\frac{1}{x}$

Short answer

The graph of $y=\frac{1}{x}$is

Step-by-step solution

Step 1. Given data

The given equation is

$y=\frac{1}{x}$

Step 2. determine intercepts

For x-intercepts, take $y=0$

$$\begin{gathered} y=\frac{1}{x} \\ 0=\frac{1}{x} \end{gathered}$$

solution of x is not possible so there are no x-intercepts

For y-intercepts, take $x=0$

$$\begin{gathered} y=\frac{1}{x} \\ y=\frac{1}{0} \end{gathered}$$

solution ofyis undefined so there are noy-intercepts.

Step 3. x-axis symmetry

To check the symmetry with respect to the x-axis, substitute$-y$for $y$in the equation

$$\begin{gathered} -y=\frac{1}{x} \\ y=-\frac{1}{x} \end{gathered}$$

the equation is not equivalent to the original equation

so the graph of the equation is not symmetric with respect to x-axis

Step 4. y-axis symmetry

To check the symmetry with respect to the y-axis, substitute$-x$ for x in the equation

$$\begin{gathered} y=\frac{1}{-x} \\ y=-\frac{1}{x} \end{gathered}$$

the equation is not equivalent to the original equation

so the graph of the equation is not symmetric with respect to y-axis

Step 5. origin symmetry

To check the symmetry with respect to the y-axisand x-axis, substitute$-x$ for x and$-y$ for yin the equation

$$\begin{gathered} -y=\frac{1}{-x} \\ y=\frac{1}{x} \end{gathered}$$

the equation is equivalent to the original equation

so the graph of the equation is symmetric with respect to the origin

Step 6. Coordinates of points of the graph

Consider the different values of x and determine coordinates of several points

Step 7. Graph

Plot the points and connect them by the smooth curve