Problem 104 - College Algebra

Question

Exercises will help you prepare for the material covered in the next section. Determine whether \(f(x)=x^{4}-2 x^{2}+1\) is even, odd, or neither. Describe the symmetry, if any, for the graph of \(f\)

Step-by-Step Solution

Answer

The function \(f(x)=x^{4}-2 x^{2}+1\) is an even function and its graph exhibits symmetry about the y-axis.

Step 1: Define the Function

We're given the function \(f(x)=x^{4}-2 x^{2}+1\).

Step 2: Determine if Function is Even

For a function to be even, it must hold true that \(f(-x) = f(x)\). So, we’ll calculate the value of \(f(-x)\).Substituting \(x\) with \(-x\) in the equation will give us:\[f(-x)=(-x)^{4}-2 (-x)^{2}+1 = x^{4}-2 x^{2}+1\]Comparing the result with \(f(x)\), we can see they are equal, hence \(f(x)\) is an even function.

Step 3: Describe the Symmetry

Since \(f(x)\) is found to be an even function, we know that it exhibits symmetry about the y-axis. In other words, the left half and the right half of the graph of \(f(x)\) mirror each other across the y-axis.

Key Concepts

Symmetry in Functions

Understanding symmetry in functions is crucial to recognizing patterns and predicting the behavior of graphs. When we consider a function like f(x) = x^4 - 2x^2 + 1, we're looking for a type of reflection. Specifically, we check for two types of symmetry: even and odd. A function is said to be even if reflecting it over the y-axis gives us the same graph. Mathematically, this means for every x, f(x) = f(-x). A function is odd if rotating it 180 degrees around the origin gives us the same graph, which implies f(-x) = -f(x) for all x.
In the case of the given function, after replacing x with -x, we find that the original function doesn't change, which confirms that the function is even and possesses y-axis symmetry. This means if you were to fold the graph along the y-axis, each point on the right side would match a corresponding point on the left.

Graphing Polynomials

Graphing polynomials can seem daunting, but understanding the structure of a polynomial function can make it much more approachable. The general form of a polynomial is P(x) = a_nx^n + ... + a_1x + a_0, where each a_i (i being any subscript) is a coefficient and n is the degree of the polynomial, which indicates the highest power of x.
For the polynomial f(x) = x^4 - 2x^2 + 1, it is a quartic polynomial (because the highest power of x is four). The graph of this function starts end-up on both ends since the leading coefficient (the coefficient of the highest power term, which is x^4 in this case) is positive. If you were to draw the graph, it would resemble a 'W' shape but with the central 'V' being more shallow due to the -2x^2 term. By identifying the turning points and symmetry, we can sketch a rough graph of this polynomial.

Function Transformation

Function transformation takes a base function and modifies it through various operations such as shifting, stretching, compressing, or reflecting. These transformations can be described by changing the function's equation, which then alters the graph's appearance. For example, adding a constant c to a function f(x) as in f(x) + c would shift the graph up by c units.
When looking at the polynomial f(x) = x^4 - 2x^2 + 1, we can break it the down into transformations of a simpler function, g(x) = x^4. First, g(x) is transformed by the term -2x^2, which changes the curve's nature, making it wider and introducing additional turning points. Lastly, adding 1 shifts the entire graph up by 1 unit. By understanding these individual transformations, we can predict the graph's shape without plotting each point.