Problem 87 - A Graphical Approach to Precalculus with Limits

Question

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=\frac{1}{4 x^{3}}$$

Step-by-Step Solution

Answer

The function is symmetric about the origin.

Step 1: Test for Symmetry About the y-axis

To check if the function is symmetric about the y-axis, replace \(x\) with \(-x\) and see if the resulting expression is the same as \(f(x)\). Calculate \(f(-x)\): \[ f(-x) = \frac{1}{4(-x)^3} = \frac{1}{-4x^3} = -\frac{1}{4x^3} \]Since \(f(-x) eq f(x)\), the function is not symmetric about the y-axis.

Step 2: Test for Symmetry About the Origin

To check if the function is symmetric about the origin, replace \(x\) with \(-x\) and check if \(f(-x) = -f(x)\). We have already found \(f(-x) = -\frac{1}{4x^3}\). Now, calculate \(-f(x)\):\[-f(x) = -\left(\frac{1}{4x^3}\right) = -\frac{1}{4x^3} \]Since \(f(-x) = -f(x)\), the function is symmetric about the origin.

Step 3: Graph the Function to Verify Symmetry

Plot the function \(f(x) = \frac{1}{4x^3}\) using a calculator. Observe the graph: It should appear the same on opposite quadrants, confirming that it is symmetric about the origin. There are no reflections over the y-axis.

Key Concepts

Graphing Functions

Graphing a function involves plotting its points on a coordinate system, represented usually by the x-axis and y-axis. The graph of a function provides a visual representation of the behavior of the function. Here are some general steps to graphing a function:

• Choose several values of x, both positive and negative, to substitute into the function.
• Calculate the corresponding y values.
• Plot the points on a Cartesian plane.
• Connect the points smoothly, following the trend indicated by the values.

When working with more complex functions, like rational or polynomial functions, using a graphing calculator or software can be extremely helpful to get an accurate picture. In the case of the function \(f(x) = \frac{1}{4x^3}\), understanding its graph helps in analyzing its symmetry properties.

Symmetry About the Origin

A function is symmetric about the origin if, for every point \((x, y)\) on the graph, the point \((-x, -y)\) is also on the graph. Mathematically, this means that \(f(-x) = -f(x)\) for every x in the domain of the function.
To determine if a graph is symmetric about the origin, follow these steps:

• Calculate \(f(-x)\) by replacing x with -x in the function.
• Compare \(f(-x)\) with \(-f(x)\).

For the function \(f(x) = \frac{1}{4x^3}\), when \(x\) is replaced with \(-x\), the result \(f(-x) = -\frac{1}{4x^3}\) matches \(-f(x)\), confirming its origin symmetry. This means the graph will look exactly the same if rotated 180 degrees around the origin.

Symmetry About the y-axis

Symmetry about the y-axis means that a function's graph reflects itself across the y-axis. For a function to have this symmetry property, the condition \(f(-x) = f(x)\) must hold true for every x in its domain.
Here’s how to test for y-axis symmetry:

• Replace x with -x in the function to find \(f(-x)\).
• Check if \(f(-x)\) is equal to \(f(x)\).

In the example of the function \(f(x) = \frac{1}{4x^3}\), substituting \(-x\) results in \(f(-x) = -\frac{1}{4x^3}\), not equal to \(f(x)\). Hence, this function does not exhibit symmetry about the y-axis. It's a useful test to quickly determine the nature of symmetry in polynomial and rational functions.