Problem 72 - 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. $$f(x)=x^{3}-3 x$$

Step-by-Step Solution

Answer

The function is symmetric with respect to the origin.

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

For a graph to be symmetric with respect to the y-axis, the function must satisfy the condition \( f(x) = f(-x) \) for all \( x \). Substitute \( -x \) into the function: \( f(-x) = (-x)^3 - 3(-x) \). Simplifying gives: \( f(-x) = -x^3 + 3x \). Since \( f(-x) = -x^3 + 3x eq f(x) \), the function is not symmetric with respect to the y-axis.

Step 2: Determine Symmetry with respect to the Origin

For a graph to be symmetric with respect to the origin, it must satisfy the condition \( f(-x) = -f(x) \) for all \( x \). We already found that \( f(-x) = -x^3 + 3x \). The negative of the original function \( f(x) = x^3 - 3x \) is \( -f(x) = -x^3 + 3x \). Both expressions are equal, therefore, the function is symmetric with respect to the origin.

Step 3: Verify with Graphing Calculator

Use a graphing calculator to plot \( f(x)=x^3-3x \) in the standard window. Observe the plot to confirm symmetry. The graph should reflect that the function is symmetric with respect to the origin, as it exhibits rotational symmetry around the origin.

Key Concepts

Symmetric with Respect to the Y-Axis

When discussing graph symmetry, one common form is symmetry with respect to the y-axis. This means that the left and right sides of the graph mirror each other across the y-axis. Mathematically, a function is symmetric with respect to the y-axis if, when replacing every instance of \(x\) with \(-x\), the resulting expression is identical to the original function. This can be expressed as:

• \( f(x) = f(-x) \) for all \( x \)

This rule tells us that flipping the graph over the y-axis does not change its appearance. In the given function \( f(x)=x^{3}-3x \), substituting \(-x\) for \(x\) results in \( f(-x) = -x^3 + 3x \). Clearly, \( f(-x) \) is not equal to \( f(x) \), indicating the graph of this function is not symmetric with respect to the y-axis. Understanding and checking this condition helps quickly identify this type of symmetry in polynomial functions, among others.

Symmetric with Respect to the Origin

Origin symmetry involves a different type of mirroring. A function is symmetric with respect to the origin if rotating the graph 180 degrees around the origin leaves it unchanged. This is sometimes called point symmetry. For a function to exhibit this symmetry, the rule is:

• \( f(-x) = -f(x) \) for all \( x \)

In simpler terms, taking the negative of both the input and the output of the function should lead back to the original function. For \( f(x) = x^3 - 3x \), by replacing \(x\) with \(-x\), we find \( f(-x) = -x^3 + 3x \) and the negative of the original function \( -f(x) = -x^3 + 3x \). As these two are identical, the function is symmetric with respect to the origin. Recognizing this symmetry is crucial when sketching graphs or analyzing their behavior.

Polynomial Function

Polynomial functions form a significant component of mathematical studies and exhibit a range of symmetrical properties based on their degree and coefficients. A polynomial function, like \( f(x) = x^3 - 3x \), is composed of terms involving powers of \( x \). In this instance, the function has terms up to \( x^3 \), making it a cubic polynomial.Polynomials can exhibit various types of symmetry:

• Even-degree terms alone typically result in y-axis symmetry.
• Odd-degree terms alone could lead to origin symmetry.

Overall, analyzing polynomial functions involves noting their highest powers and evaluating these based on well-known rules of symmetry. In our function, the mixture of odd powers led us to observe origin symmetry, a useful insight for graph analysis.

Analytic Method

The analytic method is a powerful tool for determining symmetry in mathematical functions. It provides a rigorous process for evaluating whether specific types of symmetry exist. This method involves substituting values systematically:- Substituting \(-x\) into the function \( f(x) \) to find both \( f(-x) \) and \(-f(x) \)- Comparing these results to \( f(x) \)Through these steps, you can mathematically confirm symmetry types:

• Y-axis symmetry: Check if \( f(-x) = f(x) \).
• Origin symmetry: Check if \( f(-x) = -f(x) \).

By applying this method to the function \( f(x) = x^3 - 3x \), we've shown the lack of y-axis symmetry and confirmed origin symmetry. This technique enhances understanding and allows for precise graphing without reliance solely on visual tools or graphing calculators.