Problem 79 - 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. $$f(x)=0.5 x^{4}-2 x^{2}+1$$

Step-by-Step Solution

Answer

The graph is symmetric with respect to the \(y\)-axis.

Step 1: Understand Symmetry Types

The graph of a function can have three types of symmetry: 1) Symmetric with respect to the \(y\)-axis: if \(f(-x) = f(x)\) for all \(x\). 2) Symmetric with respect to the origin: if \(f(-x) = -f(x)\) for all \(x\). 3) Neither: if neither condition holds. We will test these conditions for the function \(f(x) = 0.5x^4 - 2x^2 + 1\).

Step 2: Test for Y-axis Symmetry

Calculate \(f(-x)\) for the function and compare it to \(f(x)\). Substitute \(-x\) into the function: \(f(-x) = 0.5(-x)^4 - 2(-x)^2 + 1\). Simplify to find that \(f(-x) = 0.5x^4 - 2x^2 + 1\). Therefore, \(f(-x) = f(x)\), which indicates that the function is symmetric with respect to the \(y\)-axis.

Step 3: Test for Origin Symmetry

Calculate \(f(-x)\) and compare it to \(-f(x)\). We already found \(f(-x) = 0.5x^4 - 2x^2 + 1\) in the previous step. Now, find \(-f(x)\): \(-f(x) = -(0.5x^4 - 2x^2 + 1) = -0.5x^4 + 2x^2 - 1\). Since \(f(-x) eq -f(x)\), the function is not symmetric with respect to the origin.

Step 4: Use a Calculator

Graph \(f(x) = 0.5x^4 - 2x^2 + 1\) using a graphing calculator. Observe the graph within a standard window range (e.g., \(-10 \leq x \leq 10\), \(-10 \leq y \leq 10\)). Ensure that it reflects the algebraic finding of symmetry with respect to the \(y\)-axis.

Key Concepts

Understanding Y-axis Symmetry

Y-axis symmetry is an important concept when analyzing the symmetry of functions. A function is said to be symmetric with respect to the y-axis if the graph of the function does not change when it is reflected over the y-axis. Mathematically, this can be expressed as \( f(-x) = f(x) \) for all values of \( x \).
To determine if a function is y-axis symmetric, one should substitute \( -x \) into the function and simplify the resulting expression. If the simplified expression is equal to the original function, then the function is symmetric with respect to the y-axis.
For example, in the given function:

• \( f(x) = 0.5x^4 - 2x^2 + 1 \)
• Compute \( f(-x) = 0.5(-x)^4 - 2(-x)^2 + 1 \), which simplifies to \( 0.5x^4 - 2x^2 + 1 \).
• Since \( f(-x) = f(x) \), it confirms that the function is symmetric with respect to the y-axis.

This symmetry implies that the function's graph looks the same on both sides of the y-axis. It's like folding the graph along the y-axis and having both halves perfectly overlap. Checking for y-axis symmetry helps understand the shape and behavior of a function's graph.

Exploring Origin Symmetry

Origin symmetry, also known as rotational symmetry about the origin, occurs in a function when the graph remains unchanged when rotated 180 degrees around the origin. This type of symmetry is defined as \( f(-x) = -f(x) \) for all \( x \).
To test a function for origin symmetry, first calculate \( f(-x) \) and then compare it with \(-f(x) \). If \( f(-x) \) equals \(-f(x) \), then the function has origin symmetry. In the example of the function \( f(x) = 0.5x^4 - 2x^2 + 1 \):

• We already found \( f(-x) = 0.5x^4 - 2x^2 + 1 \).
• Compute \( -f(x) = -(0.5x^4 - 2x^2 + 1) = -0.5x^4 + 2x^2 - 1 \).
• Since \( f(-x) eq -f(x) \), the function is not symmetric with respect to the origin.

Understanding whether a function has origin symmetry enhances the prediction of its visual appearance and behavior around the origin. Functions with this symmetry can be imagined as being spun 180 degrees around the coordinate plane origin, and the graph would still look the same. In conclusion, this function does not exhibit origin symmetry, and this can be easily verified with the algebraic method described.

Using a Graphing Calculator

A graphing calculator is a potent tool for visualizing the symmetry of functions. By plotting the function's graph, one can visually confirm the algebraic findings of symmetry.
Graphing calculators offer various display features and allow users to input equations directly to see their graphical representations.
To use a graphing calculator to check symmetry, follow these steps:

• Input the function, such as \( f(x) = 0.5x^4 - 2x^2 + 1 \).
• Set a standard viewing window, commonly \( -10 \leq x \leq 10 \, -10 \leq y \leq 10 \), to get a comprehensive view of the graph.
• Observe the graph to see if it mirrors itself over the y-axis (for y-axis symmetry) or appears unchanged when rotated around the origin (for origin symmetry).

In this context, plotting the function on a graphing calculator confirms the analytic determination that the function is symmetric with respect to the y-axis. The graph will appear the same on both sides of the y-axis, enabling a clearer understanding of the function's behavior. Practicing with graphing calculators strengthens comprehension of symmetry concepts and illustrates the relationships between algebraic expressions and their visual representations.