Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Using a Graphing Utility to Graph an Equation In Exercises \(31-44,\) use a graphing utility to graph the equation. Use a standard viewing window. Approximate any \(x\) - or \(y\) -intercepts of the graph. \(y=\frac{2 x}{x-1}\)

Step-by-Step Solution

Answer

The graph of the function \(y=\frac{2x}{x-1}\) has a x-intercept at \(x=0\) and a y-intercept at \(y=-2\).

Step 1: Graphing the Equation

Enter the equation \(y=\frac{2x}{x-1}\) into a graphing utility. This will yield a visual representation of the function, which will allow one to simply read off the intercepts, if they exist. The standard viewing window works fine for this task.

Step 2: Finding the x-intercept

Set \(y=0\) and solve for \(x\). The solution to this equation will give you the x-intercept. However, for the equation \(y=\frac{2x}{x-1}\), when we set \(y=0\), we get \(0=\frac{2x}{x-1}\), which yields \(x=0\).

Step 3: Finding the y-intercept

Set \(x=0\) and solve for \(y\). The solution to this equation will give you the y-intercept. However, plugging \(x=0\) into the equation \(y=\frac{2x}{x-1}\), yields \(y=\frac{2.0}{0-1}= -2\).

Key Concepts

x-intercept

An intercept is the point where a graph crosses either the x-axis or the y-axis. To find the x-intercept, set the equation equal to zero and solve for the variable x. For example, in the equation given, \( y=\frac{2x}{x-1} \), setting \( y=0 \) gives you \[ 0 = \frac{2x}{x-1} \].
When you solve this equation by multiplying both sides by \( x-1 \), it simplifies to \( 0 = 2x \), leading to \( x=0 \).
This means the graph crosses the x-axis at the origin point: \((0,0)\).
It's essential to remember:

• The x-intercept always has a zero for the y-coordinate.
• Solving for x when y equals zero reveals the x-intercept.

y-intercept

Finding the y-intercept involves setting x to zero and solving for y. The y-intercept is a key feature of the graph as it shows where the function crosses the y-axis. Consider the equation \( y=\frac{2x}{x-1} \).
By setting \( x=0 \), we substitute zero into the equation:
\[ y = \frac{2(0)}{0-1} \].
This calculation simplifies to \( y = \frac{0}{-1} = -2 \).
Thus, the y-intercept is \((0,-2)\).
To remember:

• The y-intercept always has zero for the x-coordinate.
• Setting x to zero in the equation finds the y-intercept.

graphing utility

A graphing utility is a wonderful tool that helps visualize the behavior of functions. When you input any equation into this software, it creates a graph that you can inspect for intercepts and other features. For the equation \( y=\frac{2x}{x-1} \), using a graphing utility:

• Enables you to easily spot both the x- and y-intercepts.
• Helps identify aspects like symmetry, intercepts, and asymptotes that might not be obvious from the equation alone.

To use the graphing utility effectively:

• Enter the equation accurately into the software.
• Use a standard viewing window for an initial overview.
• Zoom in or adjust the window for a closer look at specific parts of the graph if necessary.

Overall, a graphing utility saves time and ensures accurate results when working with complex equations.

ratio functions

In mathematics, a ratio function involves dividing one expression by another, like \( y=\frac{2x}{x-1} \). Ratio functions often have features such as intercepts, asymptotes, and points of discontinuity. Here’s why they are interesting and important:

• They often have vertical asymptotes, where the function is undefined, such as \( x=1 \) in our example.
• They may have horizontal asymptotes that the graph approaches as x heads towards infinity in either direction.
• Understanding them helps with grasping broader mathematical concepts related to limits and continuity.

To work with ratio functions effectively, always:

• Check for undefined points by setting the denominator equal to zero.
• Use a graphing utility to observe the function’s overall behavior visually.
• Calculate x- and y-intercepts just like any other function by setting the opposite variable in the equation to zero.

In essence, ratio functions are fundamental in exploring real-world scenarios where relationships are proportional and can grow infinitely large or small. They introduce a deeper understanding of how equations model natural phenomena.