Question

If the graph of an equation is symmetric with respect to the origin and 13, - 42 is a point on the graph, then __________ is also a point on the graph.

Short answer

The symmetric point is (-13, 42).

Step-by-step solution

Understanding Symmetry with Respect to the Origin

When a graph is symmetric with respect to the origin, it means that for any point \(x, y\) on the graph, the point \(-x, -y\) is also on the graph. This reflects the graph across both the x-axis and the y-axis.

Identifying the Given Point

The given point on the graph is (13, -42). To find the point that is symmetric to this with respect to the origin, we need to take the opposite of both the x-coordinate and the y-coordinate.

Determining the Symmetric Point

To find the symmetric point, change the sign of both the x and y coordinates of the given point (13, -42). This results in the point (-13, 42).

Key concepts

Origin Symmetry

When we talk about a graph being symmetric with respect to the origin, it means that for any point \(x, y\) on the graph, the point \(-x, -y\) must also be on the graph. This creates a kind of mirror image across both the x-axis and the y-axis. For instance, if the point \(13, -42\) is on the graph, the point \(-13, 42\) must also be on the graph. Think of it as flipping the graph both horizontally and vertically around the origin.

Coordinates Transformation

Coordinates transformation is a key concept for understanding symmetry and other geometric properties of graphs. When we say transformation in this context, we are referring to changing the x and y coordinates of a point, to get another point. For origin symmetry:

• Change the sign of the x-coordinate
• Change the sign of the y-coordinate

This simple transformation gives you the symmetric point. So, if you start with \(13, -42\), transforming the coordinates will give you \(-13, 42\). This tells us how every point relates to its symmetric counterpart in origin symmetry.

Graph Reflection

Graph reflection is an important idea when it comes to understanding how symmetry works. Reflecting a graph means creating a mirror image. For origin symmetry, reflect each point across both the x-axis and the y-axis:

• Flip the x-coordinate to get its negative
• Flip the y-coordinate to get its negative

This reflection ensures the new point is directly opposite the original point concerning the origin. For the given point \(13, -42\), reflecting across both axes gives \(-13, 42\). This kind of symmetry is commonly used in mathematics to analyze and understand the behavior of equations and their graphs.