Problem 58 - Calculus with Concepts in Calculus

Question

Suppose the graph of an equation is symmetric with respect to the \(y\) axis and the origin. Is it necessarily symmetric with respect to the \(x\) axis? Explain.

Step-by-Step Solution

Answer

Yes, symmetry with respect to the y-axis and origin implies x-axis symmetry.

Step 1: Examine Symmetry with Respect to the Y-axis

For a graph to be symmetric with respect to the y-axis, if point \((x, y)\) is on the graph, then \((-x, y)\) must also be on the graph. This means the equation of the graph does not change if \(x\) is replaced by \(-x\).

Step 2: Examine Symmetry with Respect to the Origin

For a graph to be symmetric with respect to the origin, if point \((x, y)\) is on the graph, \((-x, -y)\) must also be on the graph. This implies that the equation remains equivalent if both \(x\) and \(y\) are replaced by their negative values.

Step 3: Determine if X-axis Symmetry is Necessary

We need to check if these symmetries necessarily imply symmetry with respect to the x-axis, where point \((x, y)\) implies the presence of \((x, -y)\). Using the transformations, if a point \((x, y)\) and \((-x, y)\) are on the graph (due to y-axis symmetry), and \((x, y)\) and \((-x, -y)\) are on the graph (due to origin symmetry), we derive \((-x, y)\) and \((x, y)\) are paired with \((x, -y)\) and \((-x, -y)\), thus also meeting the x-axis symmetry where \((x, -y)\) is mirrored with \((x, y)\).

Step 4: Conclude the Reasoning

By linking the transformations derived from the symmetries with the y-axis and origin, it is evident that every point \((x, y)\) not only has corresponding points \((-x, y)\) and \((-x, -y)\), but also \((x, -y)\). Therefore, symmetry with respect to the y-axis and the origin implies symmetry with respect to the x-axis as well.

Key Concepts

y-axis symmetry

Y-axis symmetry is an interesting concept in mathematics. It means that a graph looks the same on the left and right sides when split by the vertical line at the y-axis. This happens when, for any point

• ext{ extbf{(x, y)}} exists on the graph, the point ext{ extbf{(-x, y)}} also lies on the graph.

If you were to fold the graph about the y-axis, the halves would match perfectly. The essential condition is that the equation remains unchanged when you replace

• extbf{x} with extbf{-x}.

This form of symmetry is common in even functions, like

• parabolas ( ext{ extbf{f(x) = x²}}), where you can see that mirroring occurs over the y-axis.

origin symmetry

Origin symmetry means the graph looks the same if you rotate it 180 degrees around the origin. In simpler terms, each point and its mirror image appear as if flipped through the center of the graph. Practically, it involves the transformation

• ext{ extbf{(x, y)}} to ext{ extbf{(-x, -y)}},

meaning if one point is on the graph, its opposite is also on the graph. This concept indicates that the equation stays valid when both

• x and y are replaced by extbf{-x} and extbf{-y}, respectively.

For instance, a simple odd function like

• f(x) = x³ showcases origin symmetry, with its corresponding flips balancing perfectly around the origin.

x-axis symmetry

X-axis symmetry involves a horizontal reflection. Here, points on the graph mirror across the x-axis. That means if a point ext{ extbf{(x, y)}} is on the graph, the point ext{ extbf{(x, -y)}} will also lie on the graph. It is essential for any graph to confirm this with a simple flip.

• When x-axis symmetry exists, replacing extbf{y} with extbf{-y} does not alter the equation.

This symmetry can be observed in horizontal opening parabolas, where the graph's shape appears identical above and below the x-axis.

graph transformations

Graph transformations are changes made to the basic graph shape that allow exploration of symmetry and shifts. These shifts can be vertical, horizontal, stretches, or flips. Each transformation alters the graph's position or orientation effectively.

• For example, translations can move graphs up or down along the y-axis or left and right along the x-axis.
• Reflections swap parts of the graph over the axes.
• Stretch or compress graphs altering their wideness or narrowness.

Understanding transformations helps draw relations between these symmetries, providing a comprehensive understanding of how graphs can be manipulated, showing how symmetry offers insights into an equation's behavior and structure.