Problem 16 - Precalculus : Building Concepts and Connections

Question

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$h(x)=2 x^{4}-x^{2}+2$$

Step-by-Step Solution

Answer

The function \(h(x)=2 x^{4}-x^{2}+2\) is an even function with y-axis symmetry.

Step 1: Test the function for Even Symmetry

First, replace \(x\) with \(-x\) and simplify the expression. If the result is equal to the original function \(h(x)\), then it's an even function. So, \(h(-x) = 2(-x)^4 - (-x)^2 + 2 = 2x^4 - x^2 + 2\). The result is identical to our original function, \(h(x)\), so the function is even.

Step 2: Test the function for Odd Symmetry

Next, replace \(x\) with \(-x\) and simplify the expression. If the result is equal to the negation of the original function \(-h(x)\), then it's an odd function. But since we already determined the function is even in Step 1, we don't need to check for odd symmetry. Because a function cannot be both odd and even at the same time.

Step 3: Classify the function

Having determined that the function is not odd, and it is indeed even, we can classify the function as an even function. This means the function has y-axis symmetry.

Key Concepts

Even Function

An even function is a type of function in which the output remains unchanged when the input is replaced by its negative counterpart. In other words, if you have a function \( f(x) \), it is even if \( f(-x) = f(x) \) for all values of \( x \) in the function's domain. This characteristic means that the graph of an even function is symmetrical around the y-axis.

• To test if a function is even, substitute \( x \) with \( -x \) in the function's equation.
• Simplify the resulting expression.
• If the expression is the same as the original function, then the function is classified as even.

In the given exercise, the function \( h(x) = 2x^4 - x^2 + 2 \) was tested for even symmetry. By substituting \( -x \), the simplified form \( h(-x) = 2x^4 - x^2 + 2 \) matches \( h(x) \), confirming that it is an even function.
This means that the function maintains its shape when reflected across the y-axis, displaying y-axis symmetry.

Odd Function

Odd functions have a unique symmetry characteristic where, when the input is replaced by its negative, the output becomes the negative of the original function. In mathematical terms, a function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all values of \( x \) in its domain. This results in the graph of the function being symmetric with respect to the origin.

• An odd function exhibits rotational symmetry.
• This means 180-degree rotation around the origin will make the graph look the same.

To determine if a function is odd, you would:

• Substitute \( x \) with \( -x \) in the function's equation.
• Simplify the equation and compare it with \(-f(x)\).
• If the resulting expression equals \(-f(x)\), the function is odd.

In our specific problem for function \( h(x) \), it was deemed unnecessary to test for oddness as the function was already classified as even. Remember, a function cannot be both even and odd simultaneously.

Y-axis Symmetry

Y-axis symmetry is a characteristic of graphs where one side of the graph is a mirror image of the other side across the y-axis. This type of symmetry is specifically associated with even functions, where \( f(x) = f(-x) \).

• A graph is said to have y-axis symmetry if every point \((x, y)\) on the graph has a corresponding point \((-x, y)\).

This symmetry results in a graph that is unchanged when flipped or reflected over the y-axis, similar to how a palindromic word reads the same forwards and backwards.
Interestingly, y-axis symmetry simplifies analyzing functions visually, as you can observe that changes on one side of the y-axis affect the other side equivalently.
For example, in our exercise, since the function \( h(x) = 2x^4 - x^2 + 2 \) is even, it demonstrates perfect y-axis symmetry, meaning it looks the same on both sides of the y-axis.