Problem 40 - 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. \(y=\sqrt[3]{x+1}\)

Step-by-Step Solution

Answer

After following the above steps, by using a graphing utility, the graph generated will show the function \(y=\sqrt[3]{x+1}\) with an x-intercept at approximately \(-1\) and no y-intercept.

Step 1: Setup the Graphing Utility

Setup the graphing utility. Enter the function \(y=\sqrt[3]{x+1}\) into the graphing utility that you are using.

Step 2: Choose a Standard Viewing Window

Choose a standard viewing window to better visualize the function. This typically includes an x-range and a y-range that includes the origin (0,0), because we are asked to find the intercepts.

Step 3: Plot the Function

Plot the function. This will give you a visual representation of the function. Observe the graph to get an idea of its behavior, shape and position in regards to the x and y axis.

Step 4: Find the y-intercept

Find the y-intercept. This can be found by observing where the graph intersects the y-axis. In general, the y-intercept is the value of the function when \(x=0\). Place your cursor on the graph where it crosses the y-axis to find this value.

Step 5: Find the x-intercept

Find the x-intercept. This is where the graph intersects the x-axis. In general, the x-intercept is found by solving the equation when \(y=0\). Move your cursor over the graph where it crosses the x-axis to find this value.

Key Concepts

Graphing Utility

When we talk about a graphing utility, we are referring to a digital tool that allows you to visually represent mathematical equations, typically in the form of graphs. This could be a software program like Desmos or GeoGebra, or a function on a graphing calculator like the TI-84. To use it effectively, you input the equation you want to graph, in this case, \(y=\sqrt[3]{x+1}\), and the utility provides a graphical representation of the equation on a coordinate plane.

The ability to see the graph visually can be immensely helpful for understanding the characteristics of the equation, such as its behavior, where it increases or decreases, and its symmetry. This visual aid is particularly valuable when trying to locate crucial points like intercepts or when examining the effects of various transformations on the graphed function.

Standard Viewing Window

The standard viewing window in graphing refers to the default range for the x and y axes that graphing utilities set to display the graph of a function. It generally encompasses a range on the x-axis from \-10\ to 10 and the same for the y-axis. This standardized view allows you to see the behavior of most basic functions around the origin, which is particularly important when identifying key features such as intercepts or symmetry.

However, some functions may require adjustments to the standard viewing window to fully capture their characteristics. For example, if a function grows very quickly or has important features outside of the standard window, you may need to zoom in or out to see those aspects clearly. In the exercise \(y=\sqrt[3]{x+1}\), using a standard viewing window gives a good starting point, but be prepared to adjust the window to ensure you capture all the intercepts of the graph.

X-Intercept

The x-intercept of a graph is a point where the graph crosses the x-axis. This point indicates where the output of the function (\(y\)) is zero. To find the x-intercept algebraically, you would set the y-value of the function to zero and solve for \(x\), which may not always be straightforward for more complex functions.

Graphing utilities simplify this process. After plotting the function, you can visually inspect where the graph crosses the x-axis or use features within the utility to precisely locate the intercept(s). For the given function \(y=\sqrt[3]{x+1}\), you would look for points on the graph where \(y=0\) to approximate the x-intercept(s). Remember that some functions can have more than one x-intercept, while others may not have any.

Y-Intercept

Similarly, the y-intercept is the point where the graph of a function intersects with the y-axis. It's an essential attribute of the graph as it represents the value of the function when \(x\) is zero. To find the y-intercept, simply substitute \(x=0\) into the function and solve for \(y\).

With graphing utilities, locating the y-intercept can be as easy as tracing the graph to where it meets the y-axis. For the function \(y=\sqrt[3]{x+1}\), you'd calculate the cubic root of (0+1), which is 1, so the y-intercept is at the point (0, 1). This point is crucial for understanding how the function behaves in relation to the y-axis and often serves as a reference point for graph transformations and symmetry.