Problem 37 - 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. \(y=x \sqrt{x+3}\)

Step-by-Step Solution

Answer

After graphing the equation \(y=x\sqrt{x+3}\) and determining the x and y -intercepts, the solutions can be approximated using the graphing utility's trace or intersect function. The exact approximation will depend on the accuracy of the graph.

Step 1: Plot the given function

Enter the equation \(y=x\sqrt{x+3}\) into the graphing utility. Make sure to use a standard viewing window in order to accurately see the behavior of the function.

Step 2: Identify the x-intercepts

Once the graph is generated, identify where the graph crosses the x-axis. These points are the x-intercepts. They represent the solutions to the equation when set equal to zero.

Step 3: Identify the y-intercept

In the same graph, identify where the graph crosses the y-axis. This point is the y-intercept. It represents the solution to the equation when x = 0.

Step 4: Approximate the intercepts

Using the trace or intersect function of the graphing utility, approximate the coordinates of the x-intercepts and y-intercepts.

Key Concepts

x-intercepts

Understanding x-intercepts can be very helpful when graphing equations. The x-intercept is the point where the graph of a function crosses the x-axis. This happens when the value of the output (usually y) equals zero.
For example, if you have an equation like \(y = x \sqrt{x+3}\), you find the x-intercepts by setting \(y\) to zero and solving for \(x\). This gives the solution points where the equation crosses the x-axis.
Identifying x-intercepts helps you understand where a graph might cross or touch the horizontal axis, providing insight into the function’s behavior.

y-intercepts

Similarly, y-intercepts are crucial when trying to understand graphs. The y-intercept is where the graph intersects the y-axis. This occurs when the input, \(x\), is zero. By finding where the graph crosses the y-axis, you can determine the initial value or starting point of a function when \(x = 0\).
In our example equation \(y = x \sqrt{x+3}\), the y-intercept can be found by substituting \(x = 0\) into the equation. Calculating this will give you the point on the graph where it meets the y-axis.
Knowing both x- and y-intercepts can help effectively position the graph in the coordinate plane.

graphing utility

A graphing utility is a helpful tool when working with equations like \(y = x \sqrt{x+3}\). These digital tools depict the graphs of equations, making complex functions easier to visualize.

• They enable detailing analysis of graph features such as peaks, valleys, and intercepts.
• Provide a quick way to understand the behavior of a function over different intervals.
• Enhance efficiency by processing calculations that would otherwise be lengthy and complex.

To use a graphing utility, input the entire equation as it's presented, and these tools will plot it automatically. This visualization aids in identifying key points like x- and y-intercepts.

standard viewing window

When using a graphing utility, selecting the correct standard viewing window is essential. The standard viewing window refers to the default range of the graph which shows both the x- and y-axes clearly.

• Ensures adequate coverage of relevant points of interest on the graph.
• Makes sure intercepts and important features of the graph are visible and can be analyzed easily.
• Provides a balanced representation, preventing challenges in interpreting distorted or incomplete graphs.

When graphing \(y = x \sqrt{x+3}\), make sure the window is set to showcase the behavior of the entire function, including any pertinent intercepts. Proper adjustments might sometimes be required to capture all necessary details.