Problem 78 - 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. $$f(x)=x^{5}-2 x^{3}$$

Step-by-Step Solution

Answer

The function is symmetric with respect to the origin.

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

To determine if a function is symmetric with respect to the y-axis, check if \(f(-x) = f(x)\). Calculate \(f(-x)\) for the given function. Substitute \(-x\) into the function \(f(x) = x^5 - 2x^3\):\[ f(-x) = (-x)^5 - 2(-x)^3 = -x^5 + 2x^3.\] Since \(f(-x) eq f(x)\), the function is not symmetric with respect to the y-axis.

Step 2: Determine Symmetry with Respect to the Origin

To determine if a function is symmetric with respect to the origin, check if \(f(-x) = -f(x)\). Using the expression for \(f(-x)\) from the previous step, calculate \(-f(x)\):\[-f(x) = -(x^5 - 2x^3) = -x^5 + 2x^3.\] Since \(f(-x) = -f(x)\), the function is symmetric with respect to the origin.

Step 3: Verify with Graphical Support (Optional for Additional Confirmation)

Using a graphing calculator or software, plot the function \(f(x) = x^5 - 2x^3\) on a standard window. Observe the symmetry. The graph should appear such that for every point (x, y), there exists a point (-x, -y), confirming that it is symmetric with respect to the origin.

Key Concepts

Analytic Methods

Analytic methods in mathematics offer a structured approach to determine properties of functions like symmetry. They involve analyzing the function algebraically, instead of relying solely on visual inspection of graphs. This method is precise and removes any ambiguity that might arise from inaccurate graph plotting.
For example, if we want to explore whether a function is symmetric with respect to the y-axis or origin, we follow specific algebraic manipulations. By substituting variables and comparing expressions, we arrive at a conclusion.
So, why might this method be preferred? It doesn't rely on computational tools and provides an objective conclusion. Plus, it forms part of the foundation for further studies in calculus and algebra where precision is crucial.

Symmetry with Respect to Y-axis

Symmetry with respect to the y-axis is a key concept in understanding the graphical representation of a function. It helps identify if the graph of the function is a mirror image along the y-axis.
To check for this symmetry analytically, you replace every instance of "x" in the function with "-x" and compare the new function to the original. If the function remains unchanged, meaning: \[ f(-x) = f(x) \] then the graph is symmetric with respect to the y-axis.
This type of symmetry is only possible for certain types of functions, notably even functions. For the exercise at hand, substituting produces \(-x^5 + 2x^3\), which is not equal to the original \(x^5 - 2x^3\). Thus, it's clear the function in the exercise isn't symmetric about the y-axis.

Symmetry with Respect to Origin

Origin symmetry means that a point on a graph maps onto another such that they are reflections through the origin. This specifically applies to functions that exhibit odd properties.
To determine this symmetry using analytic methods, substitute "-x" into the function to find \(f(-x)\). Compare this result to the negative of the original function. If: \[f(-x) = -f(x)\] the graph exhibits symmetry with respect to the origin.
For example, in the given exercise, evaluating \(-f(x)\) gives us \(-x^5 + 2x^3\), identical to \(f(-x)\). Thus, the function \(x^5 - 2x^3\) is indeed symmetric with respect to the origin. This insight not only enriches our understanding of the graph but also leverages deeper comprehension of the function's nature and transformations.