Question

Graph each function. Be sure to label three points on the graph. $$f(x)=\frac{1}{x}$$

Short answer

The graphed function includes points (1, 1), (-1, -1), and (2, 0.5) and asymptotes at x=0 and y=0.

Step-by-step solution

- Identify Key Points

To graph the function, first identify key points that can be used to plot the curve. In this case, select three values for x: x = 1, x = -1, and x = 2. Then, calculate the corresponding y-values using the function: For x = 1: \[f(1) = \frac{1}{1} = 1\]For x = -1: \[f(-1) = \frac{1}{-1} = -1\]For x = 2: \[f(2) = \frac{1}{2} = 0.5\]

- Plot the Points

Plot the points (1, 1), (-1, -1), and (2, 0.5) on the coordinate plane. These points will help visualize the behavior of the function.

- Draw the Hyperbola

Draw a smooth curve through the plotted points to represent the hyperbola of the reciprocal function. Note that the graph will have vertical and horizontal asymptotes at x=0 and y=0 respectively, due to the function approaching infinity as x approaches zero.

- Label the Points and Asymptotes

Finally, clearly label the points (1, 1), (-1, -1), and (2, 0.5), as well as the vertical asymptote (x=0) and the horizontal asymptote (y=0) on the graph.

Key concepts

plotting points

When graphing reciprocal functions like \(f(x) = \frac{1}{x}\), the first step is to plot key points. Start by choosing a few values for x. For this example, we select x = 1, x = -1, and x = 2. Then, we calculate the corresponding y-values using the function:
\[f(1) = \frac{1}{1} = 1 \]
\[f(-1) = \frac{1}{-1} = -1 \]
\[f(2) = \frac{1}{2} = 0.5 \]

These calculations give us the points (1, 1), (-1, -1), and (2, 0.5) to plot on the coordinate plane. Always ensure your points are accurate, as they form the foundation of your graph.

asymptotes

Asymptotes are lines that a graph approaches but never touches. For the reciprocal function \(f(x) = \frac{1}{x}\), we have two types of asymptotes:

• Vertical Asymptote: This occurs at x = 0. As x gets closer to 0, \(f(x)\) increases or decreases without bound, making it approach infinity.
• Horizontal Asymptote: This occurs at y = 0. As x goes to infinity, both positive and negative, \(f(x)\) gets closer to 0 but never actually reaches it.

When graphing, horizontal and vertical asymptotes are crucial because they guide the shape of the hyperbola. Clearly label x=0 and y=0 on your graph to indicate the asymptotes.

hyperbolas

A hyperbola is a type of curve formed by the graph of a reciprocal function. In the case of \(f(x) = \frac{1}{x}\), the graph is a hyperbola that has two branches, one in the first and third quadrants. These branches are mirrored along the lines y = x and y = -x.

Key characteristics of hyperbolas:

• Shape: Smooth curves that move away from asymptotes.
• Quadrants: The graph of \(f(x) = \frac{1}{x}\) lies in the first and third quadrants.
• Approach: As x moves away from zero, the graph closely approaches the asymptotes but never touches them.

After plotting your points and drawing the asymptotes, draw a smooth curve through the points. Your hyperbola will extend outwards, approaching but never touching the asymptotes at x=0 and y=0.