Problem 62 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

Yes, a graph symmetric with both axes is also symmetric with respect to the origin.

Step 1: Understanding Symmetry with Respect to the X-axis

A graph is symmetric with respect to the x-axis if for every point \((x, y)\) on the graph, the point \((x, -y)\) is also on the graph. This symmetry reflects over the x-axis.

Step 2: Understanding Symmetry with Respect to the Y-axis

A graph is symmetric with respect to the y-axis if for every point \((x, y)\) on the graph, the point \((-x, y)\) is also on the graph. This symmetry reflects over the y-axis.

Step 3: Understanding Symmetry with Respect to the Origin

A graph is symmetric with respect to the origin if for every point \((x, y)\) on the graph, the point \((-x, -y)\) is also on the graph. This symmetry rotates the graph 180 degrees around the origin.

Step 4: Combining Symmetries of the X- and Y-axes

If a graph is symmetric with respect to both the x-axis and the y-axis, for a point \((x, y)\), the points \((x, -y)\) and \((-x, y)\) are both on the graph. To explore whether origin symmetry holds, check if \((-x, -y)\) is derived from these symmetries.

Step 5: Demonstrating Origin Symmetry

For a point \((x, y)\) on the graph: - Symmetry over the x-axis gives \((x, -y)\).- Symmetry over the y-axis of \((x, -y)\) gives \((-x, -y)\), which is precisely the condition for symmetry about the origin.Thus, symmetry about both axes implies origin symmetry.

Key Concepts

x-axis symmetry

A graph exhibits x-axis symmetry when it mirrors itself across the horizontal axis, the x-axis. This means that for any point often denoted \((x, y)\), there will be a corresponding point \((x, -y)\) on the graph as well. Essentially, if you fold the graph along the x-axis, each part of the graph matches perfectly with its reflected counterpart. Some key points to remember:

• Every positive y-coordinate has a matching point with a negative y-coordinate.
• The x-coordinate remains unchanged during this reflection.
• Think of an x-axis symmetry like flipping an image over a horizontal line. Your graph remains unchanged in terms of its shape, just mirrored across the line.

In practical terms, recognizing this type of symmetry simplifies analysis of the graph since you only need to consider one half of it and infer the existence of the other.

y-axis symmetry

In the case of y-axis symmetry, the graph is the mirror image of itself across the vertical axis, the y-axis. This symmetry means that for every point \((x, y)\) on the graph, there exists another point \((-x, y)\). This reflection keeps the y-coordinate constant while flipping the x-coordinate. A few things to note about y-axis symmetry:

• If you folded the graph along the y-axis, each point would line up with its mirrored counterpart.
• Even if the graph crosses the y-axis, the right half will mirror the left half.
• This symmetry is like flipping an image over a vertical line, ensuring that the graph remains in balance on either side of this axis.

Understanding this symmetry is helpful as it reduces the amount of analysis needed to understand the full behavior of the graph.

origin symmetry

Origin symmetry adds another layer of transformation by rotating the graph 180 degrees around the origin of the coordinate plane. For every point \((x, y)\) on such a graph, its counterpart must be \((-x, -y)\). This transformation essentially maps a point into the exact opposite quadrant of the plane.

• The graph looks identical even if turned upside down.
• Both coordinates of the point are negated, unlike with axis symmetries where only one coordinate changes.
• Origin symmetry implies that the graph is balanced and exhibits rotational symmetry.

Combining x-axis and y-axis symmetries leads to origin symmetry because each transformation cancels out the single reflection with a double transformation. Analyzing a graph for such symmetry can simplify tasks like sketching or interpreting the graph since this symmetry assures that every point has an equal opposite.