Graph Symmetries: Even, Odd, and Periodic Functions

Even and Odd Functions: Symmetrical Reflections

Many functions exhibit a beautiful symmetry in their graphs. The two most common types are symmetry about the y-axis and symmetry about the origin.

• An even function is symmetric with respect to the y-axis. If you were to fold the graph along the y-axis, the two halves would match perfectly.
• An odd function is symmetric with respect to the origin. This means that if you rotate the graph 180° about the origin, it will land back on itself.

The image below provides a great visual summary of these two symmetries. Notice how the parabola $y=x^2$ is a classic example of an even function, while the cubic curve $y=x^3$ exemplifies an odd function.

The Algebraic Test for Symmetry

While graphical inspection is intuitive, the definitive way to determine if a function is even or odd is through an algebraic test. These tests directly correspond to the visual symmetries we just observed.

A function $f$ is:

1. Even if $f(-x)=f(x)$ for all $x$ in its domain.
2. Odd if $f(-x)=-f(x)$ for all $x$ in its domain.

If neither of these conditions holds, the function is neither even nor odd.

Let's break down what these equations mean:

• For an even function, plugging in $-x$ gives you the exact same output as plugging in $x$. This is why the graph is a mirror image across the y-axis.
• For an odd function, plugging in $-x$ gives you the negative of the output for $x$. This creates the 180° rotational symmetry.

Your textbook provides a clear and concise explanation of these definitions.

To see these algebraic tests performed step-by-step, the following video is an excellent resource. The presenter offers a useful shortcut: for a polynomial function, if all the exponents of the variable are even, the function is even. If all the exponents are odd, the function is odd. If it's a mix, it's neither. (Note: A constant term like $c$ can be thought of as $c\cdot x^0$, and 0 is an even number).

Periodic Functions: Repeating Patterns

The third type of symmetry we'll explore is periodicity. A function is periodic if its graph repeats itself in regular intervals or cycles. Many phenomena in the natural world and in engineering, such as sound waves, AC circuits, and tidal patterns, are modeled by periodic functions.

Formally, a function $f$ is periodic if there is a positive number $T$ such that:

for all $x$ in the domain. The smallest such positive value $T$ is called the period of the function. This means the function's values repeat every $T$ units along the x-axis.

In this course, the most common periodic functions you'll encounter are the trigonometric functions.

• The functions $\sin (x)$ and $\cos (x)$ both have a period of $2\pi$.
• The function $\tan (x)$ has a period of $\pi$.

When these functions are transformed, for example $f(x)=\sin (bx)$ or $f(x)=\cos (bx)$, their period changes. The new period $T$ can be calculated with the formula:

This is a formula worth remembering, as it's fundamental for analyzing waves and oscillations.

The following video provides a concise explanation of periodic functions and demonstrates how to calculate the period.

Practice Problems

Let's test your understanding. Determine if the following functions are even, odd, or neither. For the last one, find the period.

1. $g(x)=1+3x^2-x^4$
2. $h(x)=x|x|$
3. $k(x)=\cos (3x)$

Solutions:

1. $g(x)=1+3x^2-x^4$

   • Test: $g(-x)=1+3(-x{)}^2-(-x{)}^4=1+3x^2-x^4$.
   • Since $g(-x)=g(x)$, the function is even.
   • (Shortcut: The terms are $1x^0$, $3x^2$, and $-x^4$. The exponents 0, 2, and 4 are all even.)

2. $h(x)=x|x|$

   • Test: $h(-x)=(-x)|-x|=(-x)|x|=-x|x|$.
   • Since $h(-x)=-h(x)$, the function is odd.

3. $k(x)=\cos (3x)$

   • First, let's check for even/odd. $k(-x)=\cos (3(-x))=\cos (-3x)$. A property of the cosine function is that $\cos (-\theta )=\cos (\theta )$, so $k(-x)=\cos (3x)=k(x)$. The function is even.
   • Next, let's find the period. Using the formula $T=\frac{2\pi }{|b|}$ with $b=3$, we get $T=\frac{2\pi }{3}$. The period is $2\pi /3$.

Conclusion

In this lesson, you've added another crucial layer to your understanding of function properties. By learning to identify symmetry, you can better predict a function's behavior and appreciate the underlying structure in its formula and graph.

Key Takeaways:

• Even functions are symmetric about the y-axis and satisfy the algebraic rule $f(-x)=f(x)$.
• Odd functions are symmetric about the origin (a 180° rotation) and satisfy the rule $f(-x)=-f(x)$.
• Periodic functions repeat their values at regular intervals of length $T$ (the period) and satisfy the rule $f(x+T)=f(x)$.
• For trigonometric functions like $\sin (bx)$ and $\cos (bx)$, the period is given by $T=2\pi /|b|$.

In our next lesson, we will begin exploring a "library" of essential functions. We will start by learning to identify and sketch polynomial, power, and rational functions by analyzing their key features. The skills you've developed so far—finding the domain and range and identifying symmetry—will be invaluable as we start dissecting these important function families.