Problem 84 - 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. $$f(x)=x^{3}-3 x$$

Step-by-Step Solution

Answer

The function is symmetric with respect to the origin.

Step 1: Identify the Types of Symmetry

A function can be symmetric with respect to the y-axis if replacing \(x\) with \(-x\) yields the same function, i.e., \(f(-x) = f(x)\). It is symmetric with respect to the origin if \(f(-x) = -f(x)\). We'll test these conditions.

Step 2: Calculate \(f(-x)\)

Substitute \(-x\) into the function: \(f(-x) = (-x)^3 - 3(-x) = -x^3 + 3x\).

Step 3: Test for Symmetry with Y-Axis

Compare \(f(-x)\) with \(f(x)\). Here, \(f(-x) = -x^3 + 3x\) and \(f(x) = x^3 - 3x\). Since they are not equal, the function is not symmetric with respect to the y-axis.

Step 4: Test for Symmetry with Origin

Check if \(f(-x) = -f(x)\). Here, \(-f(x) = -(x^3 - 3x) = -x^3 + 3x\), which matches \(f(-x)\). Therefore, \(f(-x) = -f(x)\) and the function is symmetric with respect to the origin.

Key Concepts

Understanding Y-Axis Symmetry

To determine if a function graph has y-axis symmetry, we check how it reacts when we substitute the variable with its negative counterpart. If replacing every instance of \(x\) with \(-x\) in the function results in the same function, then the graph is symmetric about the y-axis. This symmetry implies the graph can be folded along the y-axis, producing two mirrored halves.
For example, let's consider the function given: \(f(x) = x^3 - 3x\). We substitute \(-x\) to calculate \(f(-x)\): - \(f(-x) = (-x)^3 - 3(-x) = -x^3 + 3x\)
Upon comparison, we see that \(f(-x) eq f(x)\) because \(-x^3 + 3x\) is not equal to \(x^3 - 3x\). This confirms that this function does not possess y-axis symmetry.
Remember, having y-axis symmetry means every point \((x, y)\) on the graph has a corresponding point \((-x, y)\) as well.

Exploring Origin Symmetry

Origin symmetry is another type of symmetry found in certain functions. A function displays this type of symmetry if the graph is invariant when rotated 180 degrees about the origin. Analytical methods allow us to determine this by checking if replacing \(x\) with \(-x\) results in the negative of the entire function, mathematically expressed as \(f(-x) = -f(x)\).
Using the function we have: \(f(x) = x^3 - 3x\). Substituting \(-x\) gives us: - \(f(-x) = -x^3 + 3x\)
Now, compute \(-f(x)\): - \(-f(x) = -(x^3 - 3x) = -x^3 + 3x\)
We find that \(f(-x) = -f(x)\), which confirms that the function \(x^3 - 3x\) exhibits origin symmetry. This means that for any point \((x, y)\) on the graph, there is a corresponding point \((-x, -y)\).

Using an Analytical Method

In mathematics, using an analytical method refers to employing algebraic techniques to verify properties or determine attributes of functions. When determining symmetry, the analytical method involves substituting \(-x\) for \(x\) in the function to see if it behaves in a specific way that indicates symmetry.
Symmetry assessments fall into two main types, as seen with our function \(f(x) = x^3 - 3x\):

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

The analytical method is a powerful tool because it allows us to predict graphical behavior from purely algebraic expressions, without needing to plot them visually. It's just like using a mathematical detective toolkit to understand the deeper visual patterns in mathematical functions.