Problem 88 - 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. $$f(x)=\sqrt{x^{2}}$$

Step-by-Step Solution

Answer

The function is symmetric with respect to the y-axis.

Step 1: Understand Symmetry Types

A function has symmetry with respect to the y-axis if \( f(-x) = f(x) \) for all \( x \). It has symmetry with respect to the origin if \( f(-x) = -f(x) \) for all \( x \). If neither condition holds, the function has neither type of symmetry.

Step 2: Calculate f(-x)

Given \( f(x) = \sqrt{x^2} \), compute \( f(-x) \). \[ f(-x) = \sqrt{(-x)^2} = \sqrt{x^2} = f(x) \]

Step 3: Check y-axis Symmetry

Since \( f(-x) = f(x) \), the graph of the function is symmetric with respect to the y-axis.

Step 4: Check Origin Symmetry

For origin symmetry, check if \( f(-x) = -f(x) \).Since \( f(-x) = f(x) \) and not \( -f(x) \), the function is not symmetric with respect to the origin.

Step 5: Conclusion and Calculator Verification

Based on analytical steps, the function \( f(x) = \sqrt{x^2} \) is symmetric with respect to the y-axis and not the origin. A calculator will show a graph appearing the same on both sides of the y-axis, confirming y-axis symmetry.

Key Concepts

y-axis symmetry

When a graph shows y-axis symmetry, it means that the graph appears identical on both sides of the y-axis. This symmetry is visually observed, but to confirm y-axis symmetry analytically, we apply a transformation of the function's input. The key is to check if the function behaves the same way when we substitute negative values of the input variable. In simpler terms, a function has y-axis symmetry if replacing the input with its negative counterpart does not change the output. This translates mathematically to the condition that if for any function \( f(x) \), it holds that \( f(-x) = f(x) \) for all \( x \), then the function is symmetric with respect to the y-axis.
- For example, when considering the function \( f(x) = \sqrt{x^2} \), calculating \( f(-x) \) gives us \( \sqrt{(-x)^2} = \sqrt{x^2} \), which equals \( f(x) \). Since this mathematical relationship holds true, this confirms the function's symmetry about the y-axis.
Visualizing it with a calculator or a graphing tool will depict the function's graph looking like a mirror image on both sides of the y-axis, further confirming this characteristic.

origin symmetry

Origin symmetry in a function suggests that every coordinate point \((x, y)\) on the graph has a counterpart \((-x, -y)\) which is also on the graph. This type of symmetry can sometimes be a little harder to spot visually compared to y-axis symmetry, as it reflects around a central point rather than a line. Mathematically, a function has origin symmetry if replacing \(x\) with \(-x\) also changes the output sign, formally written as \( f(-x) = -f(x) \).
- Applying this to the same function \( f(x) = \sqrt{x^2} \), when you compute \( f(-x) \), you find it equals \( \sqrt{x^2} \), which is the same as \( f(x) \) not \(-f(x)\).
Therefore, because \( f(-x) \) does not equal \(-f(x)\), the function \( f(x) = \sqrt{x^2} \) lacks origin symmetry. In simpler terms, when you make \( x \) negative, the output doesn't become negative, so the behavior doesn't mirror across the origin.

analytical method

The analytical method is a powerful approach to determining symmetries, and it enables precise conclusion without solely relying on graphical plots. By examining specific algebraic transformations of the function, we assess whether any symmetry is present. This method involves two main checks for symmetry types:

• For y-axis symmetry, you substitute \(-x\) for \(x\) in the function and verify if the result matches the original function \( f(x) \) (i.e., \( f(-x) = f(x) \)).
• For origin symmetry, you again substitute \(-x\) for \(x\) and check if the result is negative of the original function \( -f(x) \) (i.e., \( f(-x) = -f(x) \)).

By following these steps, the analytical method not only clarifies whether a function has y-axis or origin symmetry but also solidifies the understanding of why a function behaves or doesn't behave in a symmetric way.
These checks can be easily done with basic algebra, and by using a calculator to sketch the graph, students can see how the algebraic results align with graphical representations. This methodical approach ensures that the conclusions drawn about the function’s symmetries are both mathematically and visually confirmed.