Problem 38 - 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. \(y=(6-x) \sqrt{x}\)

Step-by-Step Solution

Answer

The solution involves graphing the given equation, which can be achieved by suitable setup of the graphing tool, computing the equation, plotting the graph and finally identifying the x and y intercepts.

Step 1: Set up the graphing utility

Open up the graphing utility and make sure it is ready to accept an equation to graph. For this equation, ensure the viewing window is set to a standard setting.

Step 2: Enter the equation

Enter the equation \(y = (6-x) \sqrt{x}\) into the graphing utility.

Step 3: Plot the graph

Plot the graph of the equation. The graph will demonstrate the relationship between \(x\) and \(y\). Pay close attention to where the graph intersects the x-axis and y-axis.

Step 4: Identify the intercepts

The x-intercepts are points where the graph intersects the x-axis, i.e., where \(y=0\). Similarly, the y-intercept is the point where the graph intersects the y-axis, where \(x=0\).These can be identified visually in the graph or can even be computed precisely using the graphing tool.

Key Concepts

X-Intercepts

Understanding the concept of x-intercepts is crucial when graphing equations.
An x-intercept is a point on the graph where the curve crosses the x-axis. At this point, the value of y is zero. To find the x-intercepts for the equation \(y=(6-x)\sqrt{x}\), you would set \(y\) to zero and solve for \(x\):
\[0 = (6-x)\sqrt{x}.\]
Manipulating this equation can help in finding the value or values of \(x\) that make the equation true, which represent the x-intercepts of the graph. When using a graphing utility, these intercepts can be approximated by observing where the graph meets the x-axis. Remember that some equations may have more than one x-intercept, no x-intercept, or an x-intercept at the origin (0,0).

Y-Intercepts

Likewise, y-intercepts play a pivotal role in graphing. A y-intercept is where the graph intersects with the y-axis, meaning the x-value is zero.
For our equation \(y=(6-x)\sqrt{x}\), finding a y-intercept requires us to substitute \(x\) with zero, if it's within the domain of the function:
\[y = (6-0)\sqrt{0}\].
However, with the square root of \(x\) present, we immediately see that our y-intercept is also at the origin, since \(\sqrt{0}\) is 0. It's important to note that while every graph has a y-intercept, it's not always possible for functions to have one if x cannot be zero (as in the case with the function \(1/x\), for example). Recognizing where these intercepts are helps to understand the function's behavior.

Standard Viewing Window

A standard viewing window in a graphing utility is a preset range for the x and y axes that allows for a quick examination of the graph's general shape. This conventional window typically displays the graph across a balanced range of values, often from -10 to 10 on both axes.
Adjusting the window is sometimes necessary for a better view of important features of a graph, especially if they lie outside the standard window. The window settings offer critical insight into the behavior of the graph across different intervals. When you enter an equation like \(y=(6-x)\sqrt{x}\) in a graphing utility, beginning with the standard window can help identify whether a more detailed or expanded view is required to capture all the intercepts. In our step-by-step solution, this would be your initial vantage point to approximate the intercepts.

Graphing Equations

Graphing equations serves as a visual representation of their solutions. When you graph the equation \(y=(6-x)\sqrt{x}\), you are revealing its geometric properties.

• Ensure your graphing utility is set up with the right window settings.
• Enter the equation accurately.
• Observe the curve's behavior as it provides clues about intercepts and continuity.

By following our problem's methodical approach, first graphing and then identifying intercepts, you can make concrete observations about the equation's characteristics. Graphing helps visualize the interplay between x and y and is an indispensable tool for enhancing your understanding of algebraic concepts.