How do you sketch the graph of y = 1/x?

To sketch the graph of \( y = \frac{1}{x} \), plot key points and draw the hyperbola with asymptotes.

First, understand that \( y = \frac{1}{x} \) is a hyperbola with two branches. The graph has vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \) respectively, meaning the curve approaches but never touches these lines.

Start by plotting key points. For positive \( x \)-values, choose \( x = 1 \) and \( x = 2 \). When \( x = 1 \), \( y = 1 \); when \( x = 2 \), \( y = \frac{1}{2} \). For negative \( x \)-values, choose \( x = -1 \) and \( x = -2 \). When \( x = -1 \), \( y = -1 \); when \( x = -2 \), \( y = -\frac{1}{2} \). These points help shape the curve.

Next, draw the hyperbola. For \( x > 0 \), the curve is in the first quadrant, approaching the axes but never touching them. For \( x < 0 \), the curve is in the third quadrant, again approaching the axes but never touching them. The graph is symmetric about the origin, meaning if you rotate it 180 degrees around the origin, it looks the same.

Remember, the graph never crosses the \( x \)-axis or \( y \)-axis. As \( x \) approaches 0 from the positive side, \( y \) becomes very large. As \( x \) approaches 0 from the negative side, \( y \) becomes very large in the negative direction. This behaviour is crucial for accurately sketching the graph.