Problem 107 - Algebra and Trigonometry

Question

Exercises 105–107 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 \(f(x) = x^{4} - 2x^{2} + 1\) is even and has symmetry about the y-axis.

Step 1: Check if the function is even

Replace \(x\) with \(-x\) in the function \(f(x) = x^{4} - 2x^{2} + 1\). If the original function is obtained, then the function is even. After replacement, we get \(f(-x) = (-x)^{4} - 2(-x)^{2} + 1 = x^{4} - 2x^{2} + 1\), which is equal to the original function. Thus, the function is even.

Step 2: Check if the function is odd

Replace \(x\) with \(-x\) in the function and it should yield \(-f(x)\) if the function is odd. After replacing, we have \(f(-x) = (-x)^{4} - 2(-x)^{2} + 1 = x^{4} - 2x^{2} + 1\), which is not equal to \(-f(x)\) since \(-f(x) = -(x^{4} - 2x^{2} + 1) = -x^{4} + 2x^{2} - 1\). Thus, the function is not odd.

Step 3: Identifying the symmetry

Since the function is even, it is symmetric about the y-axis. For a function to be even, it's graph should be mirrored along the y-axis. Therefore, when the function \(f(x) = x^{4} - 2x^{2} + 1\) is graphed, it will have a symmetric shape with respect to y-axis.

Key Concepts

Symmetry in Functions

Understanding symmetry in functions can help us to easily identify and visualize their graphs. Symmetry can be seen with reference to the y-axis, the origin, or even other axes in some cases.

**Even Functions:**
A function is considered even if it remains unchanged when we replace every instance of \(x\) with \(-x\). Mathematically, this is expressed as \(f(x) = f(-x)\). Graphically, even functions have symmetry about the y-axis. They look the same on both sides of the y-axis, which means you can fold the graph along the y-axis and it will match up perfectly.

**Odd Functions:**
A function is odd if replacing \(x\) with \(-x\) gives us \(-f(x)\). This means \(f(-x) = -f(x)\). Odd functions have rotational symmetry about the origin. Rotating an odd function graph 180 degrees around the origin will leave the graph unchanged.

**Neither:**
If a function is neither even nor odd, it doesn’t exhibit any of these symmetries. It may still have other types of symmetry, but not those related to the y-axis or the origin.

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of \(x\). Each term consists of a coefficient and a power of \(x\). These functions are quite versatile and appear in many mathematical and real-world applications.

**Forms and Degrees:**
- The degree of a polynomial is the highest power of \(x\) in the expression. For example, in \(f(x) = x^4 - 2x^2 + 1\), the degree is 4.
- Polynomial functions can be constant, linear, quadratic, cubic, quartic (fourth degree), and so on, depending on their degree.

**Coefficients:**
The coefficients in a polynomial function determine the shape and position of its graph. Changing these values can alter how the graph looks without changing its overall degree.

**Behavior:**
- As \(x\) approaches infinity, the term with the highest power of \(x\) will dictate the direction of the graph.
- Even-degree polynomials behave the same way on both extremities, while odd-degree polynomials behave oppositely.

Function Graphing

Graphing a function is a powerful way to understand its properties visually. When we graph a polynomial function like \(f(x) = x^4 - 2x^2 + 1\), we observe certain behaviors and patterns that reveal more about the function.

**Basic Plotting Techniques:**
- Start by identifying key points such as intercepts, where the function crosses the x-axis and y-axis.
- Look for symmetry. As we identified that \(f(x) = x^4 - 2x^2 + 1\) is even, it will reflect across the y-axis.

**Interpreting Polynomial Graphs:**
- Polynomial graphs often have a smooth, continuous curve. They do not have sharp points or breaks.
- Even-degree polynomials like quartic functions tend to "U" shape at the ends, meaning they rise or fall similarly on both sides.

**Analyzing Changes:**
- Observing where the graph turns can help identify local maxima and minima.
- The number of turning points is at most the degree of the polynomial minus one.