Problem 106 - 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. \(f(x)=x^{4}-2 x^{2}\)

Step-by-Step Solution

Answer

The function \(f(x) = x^4 - 2x^2\) is symmetric with respect to the y-axis and has three x-intercepts: \(x = 0, -\sqrt{2}, \sqrt{2}\).

Step 1: Graph the Function

Using a graphing utility, plot the function \(f(x)=x^{4}-2 x^{2}\).

Step 2: Identify Symmetry

A function is symmetrical with respect to the y-axis if \(f(-x) = f(x)\), the origin if \(f(-x) = -f(x)\), and the x-axis if \(f(-x) = -f(x)\). In this case, replacing \(x\) with \(-x\) in the function gives us \((-x)^4 - 2(-x)^2 = x^4 - 2x^2 = f(x)\). So the graph is symmetric with respect to the y-axis. No other symmetry is observed.

Step 3: Determine the x-Intercepts

The x-intercepts are the values of \(x\) for which \(f(x) = 0\). Solving \(x^{4}-2 x^{2} = 0\) gives us \(x = 0, -\sqrt{2}, \sqrt{2}\). These are the three x-intercepts.

Key Concepts

Symmetry in Graphs

Symmetry in graphs is a fascinating characteristic that can make analyzing a function much easier. When we speak of symmetry, we are referring to how a graph looks when mirrored over certain axes or points. There are three main types of symmetry we look for in a graph: symmetry with respect to the x-axis, the y-axis, and the origin.

- **Y-axis Symmetry**: A graph has y-axis symmetry if it looks identical when reflected across the y-axis. This occurs when for every point \(x, y\) on the graph, there is also the point \(-x, y\). In simpler terms, the left side of the graph looks the same as the right side. For a function, this means \(f(-x) = f(x)\).- **X-axis Symmetry**: This symmetry means the graph mirrors itself across the x-axis. For every point \(x, y\), there is a corresponding point \(x, -y\) on the graph.- **Origin Symmetry**: A graph is symmetric about the origin if rotating it 180 degrees about the origin retains its shape. This symmetry implies that for every point \(x, y\), there is a point \(-x, -y\) on the graph, meaning \(f(-x) = -f(x)\).
For the function \(f(x) = x^4 - 2x^2\), we substitute \(-x\) for \(x\) and find that the function remains unchanged, \(f(-x) = x^4 - 2x^2 = f(x)\), indicating symmetry with respect to the y-axis.

X-intercepts

X-intercepts are points where the graph of a function crosses the x-axis. At these points, the output (y-value) of the function is zero. Finding these intercepts is crucial because they provide important information about the function, including its roots or solutions.

To find the x-intercepts of \(f(x) = x^4 - 2x^2\), we set the function equal to zero: \((x^4 - 2x^2 = 0)\).
Next, factor the expression:

• First, factor out the common term, \(x^2\), from the equation: \(x^2(x^2 - 2) = 0\).
• Setting each factor equal to zero gives the solutions: \(x^2 = 0\) indicates \(x = 0\), and \(x^2 - 2 = 0\) gives \(x = \pm \sqrt{2}\).

Thus, the x-intercepts are \(x = 0, \pm \sqrt{2}\). These points tell us that the graph crosses the x-axis at three places, providing insight into the function's shape.

Graphing Utilities

Graphing utilities are powerful tools for visualizing mathematical functions. These tools can be anything from graphing calculators to computer software that allows you to input a function and instantly view its graph. This can be immensely helpful not only for understanding complex functions but also for identifying key features of the graph such as symmetry, intercepts, and behavior over different intervals.

Using a graphing utility often involves inputting the function as an equation and adjusting the view window to ensure all important details of the graph are visible. For the function \(f(x) = x^4 - 2x^2\), a graphing utility quickly shows the symmetric nature of the graph, as well as the points where it meets the x-axis.
Some common features of graphing utilities include the ability to:

• Zoom in and out to better view different sections of the graph.
• Calculate and display intercepts, maxima, and minima.
• Draw multiple graphs on the same set of axes to compare functions.

By allowing you to manipulate and explore functions visually, graphing utilities can deepen understanding and provide a clear view of abstract mathematical concepts.