Problem 101 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(f(x)=x^{2}(x+6)\)

Step-by-Step Solution

Answer

The function \(f(x)=x^{2}(x+6)\) has no symmetry with respect to the \(x\)-axis, \(y\)-axis, or the origin. The function has two \(x\)-intercepts, occurring at \(x=0\) and \(x=-6\).

Step 1: Graph the Function

First, enter the function \(f(x)=x^{2}(x+6)\) into a graphing utility to obtain its graphical representation. This will give an idea about its overall behavior, its points of intersections with axes and its symmetry.

Step 2: Identify Symmetry

Check for the symmetry of the function. A function is symmetric with respect to the \(y\)-axis if \(f(x)=f(-x)\). It's symmetric with respect to the \(x\)-axis if \(f(x)=-f(x)\), and it’s symmetric with respect to the origin if \(f(-x)=-f(x)\). In this function, it's noticible there is no such symmetry; the function isn't symmetric with respect to the \(x\)-axis, \(y\)-axis, or the origin.

Step 3: Determine the Number of X-Intercepts

The function intercepts the \(x\)-axis where \(f(x)=0\). Thus we equate \(f(x)=x^{2}(x+6)\) to zero and solve for \(x\). It gives 0 and -6 as the solutions. Therefore, the function has two \(x\)-intercepts, at \(x=0\) and \(x=-6\).

Key Concepts

Symmetry in Graphs

Symmetry in the context of quadratic functions, like the one given by f(x)=x^{2}(x+6), is a fascinating concept that can simplify both graphing and understanding of the function's nature. Symmetry refers to a situation where two halves of a graph mirror each other. There are generally three types of symmetry to look for in the graph of a function: symmetry with respect to the y-axis, x-axis, or the origin.

For symmetry with respect to the y-axis, we check if f(x) = f(-x), meaning that for every point (x, y) on the graph, there is a corresponding point (-x, y). If a function is symmetric with respect to the x-axis, then f(x) = -f(x), implying the graph can be flipped over the x-axis without changing the graph's shape. Last but not least, symmetry with respect to the origin is present if f(-x) = -f(x), meaning for every point (x, y), there is a point (-x, -y).

In the case of f(x)=x^{2}(x+6), the function is not exactly symmetric around any of these axes. However, all quadratic functions are symmetric with respect to a vertical line that passes through their vertex (known as the axis of symmetry), which is not necessarily the y-axis. Understanding symmetry can greatly ease the task of graphing because it can reduce the amount of work needed. If a function is symmetric with respect to an axis, you only need to plot half of it and reflect it across that axis to obtain the full graph.

X-Intercepts of a Function

The x-intercepts of a function are points where the graph of the function crosses or touches the x-axis. To find the x-intercepts algebraically, we set the function equal to zero and solve for x. This is because the y-value of any point on the x-axis is zero, reflecting the general functional notation of (x, 0).

For the quadratic function f(x) = x^{2}(x+6), finding the x-intercepts involves setting this function equal to zero and solving for x: 0 = x^{2}(x+6). This equation factors nicely, giving you x = 0 and x = -6, indicating that there are x-intercepts at these two values of x.

Graphically, these intercepts are the points where the parabola meets or intersects the x-axis, and they are key in understanding the function's root structure. Knowing the x-intercepts can also aid in sketching the graph without the need for a graphing utility. They are especially useful in many fields of study, such as physics and engineering, where they might signify points of equilibrium, rest, or change.

Using Graphing Utilities

In the digital age, graphing utilities are invaluable tools for visualizing mathematical functions, particularly when dealing with complex forms such as quadratic functions. They help in plotting accurate graphs quickly, which is useful for identifying key features like intercepts, turning points, and overall shape of the graph.

To graph f(x)=x^{2}(x+6) using a graphing utility, you simply input the function into the software or calculator. Modern graphing utilities allow for various manipulations: you can zoom in or out to get different perspectives, adjust the window to view significant points like intercepts and maximum or minimum values, and evaluate the function at specific points.

Benefits of Graphing Utilities

• Visualization: Helps to visualize the function's graph, which is crucial for understanding complex relationships in a function.
• Accuracy: Provides precise plots, reducing the risk of manual error.
• Speed: Generates graphs swiftly, saving time for analysis.
• Analysis: Many graphing utilities come with features that enable the analysis of graphs, such as tracing over a graph to read off coordinates, finding x and y-intercepts, and calculating slopes or areas under curves.

Whether you’re doing homework, preparing for an exam, or conducting research, graphing utilities are a convenient way to explore and interpret the properties of functions with minimal hassle.