Problem 75 - A Graphical Approach to College Algebra

Question

Use the analytic 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 your calculator and the standard window to support your conclusion. $$f(x)=\frac{1}{4 x^{3}}$$

Step-by-Step Solution

Answer

The function \( f(x) = \frac{1}{4x^3} \) is symmetric with respect to the origin.

Step 1: Check Symmetry with Respect to the Y-axis

To determine if a function is symmetric with respect to the y-axis, we check if \( f(-x) = f(x) \) for all \( x \).\ Start by substituting \( -x \) into the function:\[ f(-x) = \frac{1}{4(-x)^3} = \frac{1}{-4x^3} = -\frac{1}{4x^3} \]Since \( f(-x) eq f(x) \), the function is not symmetric with respect to the y-axis.

Step 2: Check Symmetry with Respect to the Origin

To determine if a function is symmetric with respect to the origin, we verify if \( f(-x) = -f(x) \) for all \( x \).\ Given that \( f(x) = \frac{1}{4x^3} \), substitute \( -x \):\[ f(-x) = -\frac{1}{4x^3} \]Compare this with \( -f(x) \):\[ -f(x) = -\left(\frac{1}{4x^3}\right) = -\frac{1}{4x^3} \]Since \( f(-x) = -f(x) \), the function is symmetric with respect to the origin.

Step 3: Visual Confirmation Using a Calculator

Use a graphing calculator to plot \( f(x) = \frac{1}{4x^3} \). Check the graph in the standard window settings.The graph should appear as a symmetrical curve regarding the origin, reinforcing the algebraic conclusion from Step 2.

Key Concepts

y-axis symmetry

Graphical symmetry often involves using the y-axis as a line of reflection. A graph shows y-axis symmetry if it mirrors itself on either side of the y-axis. This means that for every point

• To test if a function, say \( f(x) \), has this symmetry, we substitute \(-x\) for \(x\) and check if \( f(-x) = f(x) \).
• If they are equal, it will confirm that the graph looks the same from either side when flipped over the y-axis.

In our given function, \( f(x) = \frac{1}{4x^3} \), when substituting \(-x\), we get \( f(-x) = -\frac{1}{4x^3}\). Because this is not equal to \( f(x) \), we deduce it lacks y-axis symmetry.
Hence, visually, any graph with such a function will not exhibit the mirrored distribution about the y-axis.

origin symmetry

Origin symmetry implies that if you rotate the graph 180 degrees around the origin, it should look the same. This can be tested by checking if \( f(-x) = -f(x) \).
It essentially means flipping the graph twice: first over the y-axis and then over the x-axis, resulting in similar shapes.

• For our function \( f(x) = \frac{1}{4x^3} \), substituting \(-x\) yields \( f(-x) = -\frac{1}{4x^3} \).
• Performing \(-f(x)\) calculation gives \(-\left(\frac{1}{4x^3}\right) = -\frac{1}{4x^3}\).

Since \( f(-x) = -f(x) \), the function has origin symmetry.
This showcases that around the center point (origin), the graph maintains consistent visual features with this 180-degree rotational perspective.

graphing calculator

Using a graphing calculator can greatly assist in visually verifying symmetry.

• With modern calculators, you simply input \( f(x) = \frac{1}{4x^3} \) and utilize standard window settings.
• As you plot the function, observe how it demonstrates the symmetry previously discussed.
• In particular, for origin symmetry, you should see that the graph reflects itself around the origin.

A graphing calculator provides real-time visual validation of algebraic conclusions, such as seeing equal curves touching the origin for origin symmetry. Notably, this approach of plotting helps solidify your understanding of symmetry concepts by delivering immediate visual aids and comparisons.