Problem 53 - Algebra and Trigonometry

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

The graph of the equation \(y=\frac{1}{x}\) plotted for the given x-values will be a hyperbola where the x-axis acts as a vertical asymptote and the origin is the center of the curve.

Step 1: Calculate y-values

For each value of x, calculate the corresponding y-value using the equation \(y=\frac{1}{x}\). For instance, for \(x=-2\), y would be \(-\frac{1}{2}\).

Step 2: List All Points

After finding all y-values respectively, list down all the points in terms of (x, y). The points will be : (-2, -0.5), (-1, -1), (-0.5, -2), (-0.33, -3), (0.33, 3), (0.5, 2), (1,1), (2, 0.5).

Step 3: Plot the Points on the Graph

Start by drawing an x and y axis. Then plot each point on the graph. Make sure to label each point with its corresponding x, y coordinate.

Step 4: Draw the Curve

Based on the plotted points, draw a curve. As this is a rational function, it will be a hyperbola opening upwards and downwards as the x-values increase and decrease respectively.

Key Concepts

Graphing Equations

Graphing equations allows us to visualize mathematical relationships between variables. Take the equation \( y = \frac{1}{x} \) as an example. Here, \( y \) is dependent on the value of \( x \). To graph this, start by creating a table of values using selected \( x \)-values.

• Choose a range of \( x \)-values such as -2, -1, -0.5, -0.33, 0.33, 0.5, 1, and 2.
• Calculate \( y \) for each \( x \) using the equation.
• For example, if \( x = -2 \), then \( y = -0.5 \).

After doing this for all chosen values, plot these coordinate points on a graph. The process of graphing allows us to see the shape and behavior of the equation visually.

Hyperbolas

The equation \( y = \frac{1}{x} \) represents a hyperbola, which is a type of curve that appears as two separate curves. This happens because the values of \( x \) are not continuous for a rational function like this. As \( x \) gets very large or very small in either direction, \( y \) approaches zero but never actually reaches it.A hyperbola's distinguishing features include:

• Two branches: one in the top right and one in the bottom left of the Cartesian plane.
• An asymptote at the \( y \)-axis (\( x = 0 \)) because the function \( y = \frac{1}{x} \) is undefined for \( x = 0 \).

Understanding hyperbolas is essential as they frequently appear in various fields such as physics and engineering.

Coordinate Plotting

Coordinate plotting is a fundamental skill for graphing equations. It involves placing points on a graph using pairs of numbers called coordinates, which indicate a specific position. For our equation \( y = \frac{1}{x} \), coordinate plotting will help us visualize what the function looks like.Steps for plotting coordinates effectively:

• Ensure you have a coordinate plane with an \( x \)-axis and a \( y \)-axis.
• Each point on the graph is represented by a pair of values \((x, y)\).
• Plot each point by finding the intersection of its \( x \)-value and \( y \)-value.

Once plotted, connecting these points with a smooth curve provides insight into the function's behavior. Practice with various functions to improve your coordinate plotting skills.