What makes a function symmetric about the Y axis?

A function is symmetric about the Y-axis if its graph appears identical on both sides of the Y-axis, essentially acting as a mirror. This fundamental property simplifies analysis and reveals important characteristics of the function.

Understanding Y-axis Symmetry

A function that is symmetric about the Y-axis is also known as an even function. This classification is based on a specific mathematical condition that holds true for all points in its domain.

The Algebraic Condition

Y-axis symmetry occurs if "x" is replaced with "-x", and it yields the original equation. In other words, for a function $f(x)$, if you substitute $-x$ for every $x$ in the function's expression, the result is the same as the original function. This can be expressed as:

$f(-x) = f(x)$

for all values of $x$ in the function's domain.

Graphical Interpretation

From a graphical perspective, Y-axis symmetry means that if a point $(x, y)$ exists on the graph of the function, then the point $(-x, y)$ must also exist on the graph. The Y-axis serves as a line of reflection, perfectly mirroring the portion of the graph to its right onto its left, and vice-versa.

How to Test for Y-axis Symmetry

To determine if a given function is symmetric about the Y-axis, follow these steps:

1. Start with the original function: Let the function be represented as $y = f(x)$.
2. Substitute $-x$ for every $x$: In the function's expression, replace every instance of $x$ with $-x$. This will give you $f(-x)$.
3. Simplify the new expression: Perform any algebraic operations necessary to simplify $f(-x)$.
4. Compare with the original function: If the simplified $f(-x)$ is identical to the original $f(x)$, then the function is symmetric about the Y-axis (it is an even function). If $f(-x)$ is not equal to $f(x)$, it is not symmetric about the Y-axis.

Examples of Y-axis Symmetric Functions (Even Functions)

Many common functions exhibit Y-axis symmetry. Here are a few illustrative examples:

Example 1: Quadratic Function

Consider the function $f(x) = x^2$.

1. Original function: $f(x) = x^2$
2. Substitute $-x$: $f(-x) = (-x)^2$
3. Simplify: $f(-x) = x^2$
4. Compare: Since $f(-x) = x^2 = f(x)$, the function $f(x) = x^2$ is symmetric about the Y-axis. Its graph is a parabola opening upwards, clearly mirroring itself across the Y-axis.

Example 2: Absolute Value Function

Consider the function $f(x) = |x|$.

1. Original function: $f(x) = |x|$
2. Substitute $-x$: $f(-x) = |-x|$
3. Simplify: By definition of absolute value, $|-x| = |x|$. So, $f(-x) = |x|$.
4. Compare: Since $f(-x) = |x| = f(x)$, the function $f(x) = |x|$ is symmetric about the Y-axis. Its graph forms a V-shape with the vertex at the origin.

Example 3: Cosine Function

Consider the trigonometric function $f(x) = \cos(x)$.

1. Original function: $f(x) = \cos(x)$
2. Substitute $-x$: $f(-x) = \cos(-x)$
3. Simplify: From trigonometric identities, $\cos(-x) = \cos(x)$. So, $f(-x) = \cos(x)$.
4. Compare: Since $f(-x) = \cos(x) = f(x)$, the function $f(x) = \cos(x)$ is symmetric about the Y-axis. The wave-like graph of the cosine function visibly reflects across the Y-axis.

Practical Implications

Recognizing Y-axis symmetry can be incredibly useful in various mathematical and scientific contexts. It allows for:

• Simplified Analysis: When a function is known to be even, its behavior on the positive x-axis dictates its behavior on the negative x-axis, reducing the amount of work needed for analysis or plotting.
• Calculus Applications: In calculus, integrating an even function over a symmetric interval (e.g., from $-a$ to $a$) can be simplified, as $\int{-a}^{a} f(x) dx = 2 \int{0}^{a} f(x) dx$.
• Problem Solving: Understanding symmetry helps in predicting function properties and solving related problems more efficiently.

Summary of Y-axis Symmetry

Here's a quick overview of the key aspects of Y-axis symmetry: