Problem 63 - Algebra for College Students

Question

Is a graph symmetric with respect to both axes if it is symmetric with respect to the origin? Defend your answer.

Step-by-Step Solution

Answer

No, symmetry with respect to the origin does not imply symmetry with respect to the axes.

Step 1: Understand Symmetry to the Origin

A graph is symmetric with respect to the origin if for every point \((a, b)\) on the graph, the point \((-a, -b)\) is also on the graph. This implies a kind of rotational symmetry centered at the origin.

Step 2: Define Symmetry with Respect to Each Axis

A graph is symmetric with respect to the x-axis if for every point \((a, b)\) on the graph, the point \((a, -b)\) is also on the graph. Similarly, it is symmetric with respect to the y-axis if for every point \((a, b)\), the point \((-a, b)\) is also on the graph.

Step 3: Compare Symmetries

To determine if a graph symmetric with respect to the origin must also be symmetric with respect to both axes, consider a function like \(f(x, y) = xy\)). It is symmetric with respect to the origin, since \(f(a, b) = ab\) and \(f(-a, -b) = (-a)(-b) = ab\). However, it is neither symmetric with respect to the x-axis nor the y-axis, since \(f(a, -b) = -ab\) and \(f(-a, b) = -ab\).

Step 4: Conclude the Relationship

The example in Step 3 shows that a graph being symmetric with respect to the origin does not require it to be symmetric with respect to either the x-axis or the y-axis. In other words, symmetry with respect to the origin does not imply symmetry with respect to both axes.

Key Concepts

Symmetric with respect to origin

Symmetry with respect to the origin is a fascinating concept in graph theory. It occurs when a graph exhibits a type of rotational symmetry around the origin point \((0, 0)\). To be precise, for a graph to be symmetric with respect to the origin, if a point \((a, b)\) is present on the graph, the point \((-a, -b)\) must also exist.

This implies that every feature of the graph rotates into itself when spun 180 degrees around the origin. Think of it as flipping both the x and y coordinates. This characteristic makes graphs that are symmetric with respect to the origin unique.

For example, the graph of the function \(f(x, y) = xy\) is symmetric about the origin because \(f(a, b) = ab\) and \(f(-a, -b) = ab\). This dictates that the function remains unchanged when both the inputs' signs are inverted.

Symmetric with respect to x-axis

A graph is symmetric with respect to the x-axis when for every point \((a, b)\) on the graph, the point \((a, -b)\) is also on the graph. This means the graph mirrors itself across the x-axis. In simpler terms, any figure or curve above the x-axis will have an identical image below it.

Visualize it like this: Imagine you have a piece of paper with a graph drawn on it, fold the paper along the x-axis, and if the graph lays perfectly on top of itself, it is symmetric with respect to the x-axis. This kind of symmetry generally arises in even functions with swaps in the sign of the y-coordinate while keeping the x-coordinate constant.

It's important to note that a graph symmetric about the origin does not imply x-axis symmetry. For instance, consider the function \(f(x) = x^3\). It is symmetric about the origin but lacks x-axis symmetry since inverting \(b\) would change the function's value drastically.

Symmetric with respect to y-axis

Y-axis symmetry is another intriguing concept, characterized by a graph reflecting across the y-axis. Essentially, if a function exhibits this symmetry, for every point \((a, b)\) on a graph, there should be a point \((-a, b)\) as well.

This reflection means the graph's left and right sides are mirror images. Imagine placing a mirror along the y-axis, the left side of the graph appearing exactly like the right side indicates y-axis symmetry.

It's vital to differentiate that even if a graph is symmetric with respect to the origin, it doesn't necessarily guarantee y-axis symmetry. For instance, the previously mentioned function \(f(x, y) = xy\) doesn’t comply with y-axis symmetry because altering the sign of the x-coordinate causes a change in the function's value.