Problem 83 - 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. $$f(x)=x^{6}-4 x^{3}$$

Step-by-Step Solution

Answer

The graph is neither symmetric with respect to the y-axis nor the origin.

Step 1: Check Symmetry with respect to the y-axis

A function is symmetric with respect to the y-axis if \( f(-x) = f(x) \) for all \( x \). Substitute \( -x \) into \( f(x) \).\[f(-x) = (-x)^6 - 4(-x)^3 = x^6 + 4x^3\] 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

A function is symmetric with respect to the origin if \( f(-x) = -f(x) \) for all \( x \). Check this condition with the derived \( f(-x) \) from the previous step:\[f(-x) = x^6 + 4x^3\]\[-f(x) = -(x^6 - 4x^3) = -x^6 + 4x^3\] Since \( f(-x) eq -f(x) \), the function is not symmetric with respect to the origin.

Step 3: Conclusion

The function \( f(x) = x^6 - 4x^3 \) is neither symmetric with respect to the y-axis nor the origin, as proven in the previous steps.

Key Concepts

y-axis symmetry

To determine if a graph shows symmetry with respect to the y-axis, we examine if the given function remains unchanged when we replace every instance of \( x \) with \( -x \). This means we check the condition \( f(-x) = f(x) \) for every value of \( x \).

For the function \( f(x) = x^6 - 4x^3 \), substituting \( -x \) gives us \( f(-x) = x^6 + 4x^3 \). However, \( f(x) = x^6 - 4x^3 \). Clearly, \( f(-x) eq f(x) \), which shows that \( f(x) \) is not symmetric about the y-axis.

Y-axis symmetry makes the left half of the graph look identical to the right half.

• Examples of such graphs include even functions like \( f(x) = x^2 \).
• Visually, this means if you fold the graph along the y-axis, both halves would match perfectly.

origin symmetry

When a graph is symmetric with respect to the origin, every part of the graph on one side of the origin has a mirror image directly opposite to it through the origin. Mathematically, for a function to exhibit this symmetry, \( f(-x) = -f(x) \) should hold for every \( x \).

Let's apply this test again for \( f(x) = x^6 - 4x^3 \). From previous calculations, we have \( f(-x) = x^6 + 4x^3 \) and \(-f(x) = -x^6 + 4x^3 \). Since \( f(-x) eq -f(x) \), the function does not exhibit symmetry about the origin.

Graphs that display origin symmetry look unchanged even if rotated 180 degrees around the origin.

• Odd functions like \( f(x) = x^3 \) typically display this symmetry.
• The symmetry means visually the graph balances around the origin like a seesaw.

graph analysis

Graph analysis is a crucial skill for understanding the behavior and characteristics of functions. It involves examining various properties, including symmetry, to better analyze and interpret data represented graphically.

In our problem, we focused on verifying two types of symmetries: y-axis and origin. Although the function \( f(x) = x^6 - 4x^3 \) displayed neither, understanding these properties allows better graph comprehension.

Performing a graph analysis can also involve:

• Checking for asymptotes, where the graph approaches a line but never touches.
• Finding intercepts, points where the graph crosses the axes.
• Determining increasing and decreasing intervals, which highlight where the function value rises or falls.
• Identifying maximum and minimum points, giving insight into the peaks and troughs of the graph.

This detailed analysis aids not only in graph sketching but also in predictive modeling and understanding the broader function behavior.