Question

Multiple Choice To test whether the graph of an equation is symmetric with respect to the origin, replace __________ in the equation and simplify. If an equivalent equation results, then the graph is symmetric with respect to the origin. \ (a) \(x\) by \(-x\) (b) \(y\) by \(-y\) (c) \(x\) by \(-x\) and \(y\) by \(-y\) (d) \(x\) by \(-y\) and \(y\) by \(-x\)

Short answer

(c) Replace both \(x\) and \(-x\), and \(y\) with \(-y\)

Step-by-step solution

Understanding Symmetry with Respect to the Origin

For a graph to be symmetric with respect to the origin, it means that if a point \((x, y)\) exists on the graph, then the point \((-x, -y)\) must also exist on the graph. This involves changing both the x-coordinate and the y-coordinate to their opposites.

Apply Changes to the Equation

Replace every occurrence of \(x\) with \(-x\) and \(y\) with \(-y\) in the equation.

Check for Equivalence

Simplify the resulting equation. If the simplified equation is identical to the original equation, then the graph is symmetric with respect to the origin.

Correct Answer Identification

According to our understanding and steps outlined, the correct choice must replace both \(x\) and \(y\) with their negatives.

Key concepts

graph symmetry

Graph symmetry is an important concept in understanding the visual properties of equations. There are different types of symmetries such as symmetry about the x-axis, symmetry about the y-axis, and symmetry about the origin. For symmetry with respect to the origin, if a point \((x, y)\) is on the graph, then the point \((-x, -y)\) must also be present. This means that the graph looks the same if it's rotated 180 degrees around the origin. To test this symmetry, we replace every \(x\) with \(-x\) and every \(y\) with \(-y\) in the equation and check if the resulting equation is the same as the original.

equation transformation

The process of checking for symmetry involves transforming the equation. To test for origin symmetry, we perform the following steps:

• Replace every \(x\) in the equation with \(-x\).
• Replace every \(y\) in the equation with \(-y\).

Let's say we start with an equation like \(y = x^2 + 1\). After transforming, we get \(-y = (-x)^2 + 1\). Simplifying this, we find \(-y = x^2 + 1\). This transformed equation is not the same as the original, meaning the graph is not symmetric with respect to the origin. You must always simplify and compare with the original to confirm.

coordinate changes

Making coordinate changes is a crucial part of testing symmetry. Here, we swap \(x\) for \(-x\) and \(y\) for \(-y\). This flips the coordinates over both the x-axis and y-axis, effectively rotating them by 180 degrees around the origin. Understanding this, if you have the equation \(x^2 + y^2 = 1\), and you replace \(x\) with \(-x\) and \(-y\), it remains the same: \((-x)^2 + (-y)^2 = 1\). As the squares of the negatives are equal to the squares of the positives, the equation retains its original form under these changes.

algebra

Algebra is the mathematical study that helps us simplify and manipulate equations to check for symmetry. When you replace \(-x\) and \(-y\), you apply your algebraic skills to simplify the expression. Your goal is to determine if the new equation is algebraically equivalent to the original. For example, consider \( y = x^3\). Apply the changes: \(-y = (-x)^3\) simplifies to \(-y = -x^3\). This new equation \(-y = -x^3\) does match the form of the original \(y = x^3\) when multiplied by -1, confirming symmetry. Mastery of algebra allows you to navigate through these transformations and verifications with confidence.