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Chapter 0: Problem 73

Symmetry A graph is symmetric with respect to one axis and to the origin. Is the graph also symmetric with respect to the other axis? Explain.

Short Answer

Expert verified
Yes, the graph is symmetric with respect to the other axis. This conclusion comes from the understanding that a graph symmetric about the origin is symmetric about both axes.

Step by step solution

01

Understand symmetry with respect to the origin

When a graph is symmetric with respect to the origin, it implies that for every point (x, y) on the graph, there is another point (-x, -y). Essentially, if you can rotate the graph 180 degrees about the origin and the graph remains unchanged, it is symmetric about the origin.
02

Understand symmetry with respect to one axis

When a graph is symmetric with respect to one axis, it means that every point (x, y) on one side of that axis has a corresponding point on the opposite side. For instance, if a graph is symmetric with respect to the y-axis, for every point (x, y), there is another point (-x, y). If it's symmetric with respect to the x-axis, for every point (x, y), there's another point (x, -y).
03

Determine symmetry with respect to the other axis

Given that the graph is symmetric with respect to the origin and one axis, it must also be symmetric with respect to the other axis. This is because symmetry about the origin combines symmetry about the x and y axes. Given the graph is symmetric about one axis and the origin, we can conclude it is symmetric about the other axis too.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Symmetry About the Origin
Understanding symmetry about the origin is crucial in determining the characteristics of a graph. When a graph exhibits this kind of symmetry, it implies a certain balance. If you take the graph and rotate it 180 degrees around the origin, the graph will look exactly the same as it did before. This means that every point \((x, y)\)on the graph has a corresponding point\((-x, -y)\).For example, if \((2, 3)\)is a point on the graph, then \((-2, -3)\)should also be a point. This type of symmetry is quite straightforward:
  • The graph looks the same upside down.
  • Every segment of the graph mirrors along both axes simultaneously.
X-axis Symmetry
Graphs exhibiting x-axis symmetry have pairs of points across the x-axis maintaining symmetry. This means if there is a point \((x, y)\)in the graph, there will also be a point \((x, -y)\).If you fold the graph along the x-axis, the two parts should coincide. X-axis symmetry is less common in function graphs but can be found in various shapes. Key characteristics include:
  • Mirroring effect along the x-axis.
  • Upper half of the graph reflects directly downward.
Such symmetry often arises in cases where the relation or shape does not fulfill the criteria of a function, as the standard function definition does not allow a single x-value to map to multiple y-values.
Y-axis Symmetry
Y-axis symmetry is when the graph mirrors whenever you fold it along the y-axis. For each point \((x, y)\)on the graph, there is a corresponding point \((-x, y)\).This symmetry is easier to identify, as many common functions display it鈥攍ike quadratics of the form \(f(x) = x^2\).Some essential points for y-axis symmetry include:
  • The left and right sides of the graph are mirror images.
  • Commonly seen in even functions.
Y-axis symmetry ensures that the graph's form, structures, or patterns repeat themselves as you move horizontally along the y-axis, providing a balanced visual appeal.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a fundamental method of geometry where the position of points on a plane is described using an ordered pair \((x, y)\).Symmetry in this sense becomes significant, as it provides insights into the geometric properties and characteristics of shapes or graphs. Understanding coordinate geometry helps in
  • Analyzing the position and movement of graphical elements.
  • Studying complex shapes and curves analytically.
When studying symmetry in graphs, coordinate geometry assists in calculating distances, gradients, and understanding the contours and extremities based on algebraic equations. It's the bridge between algebra and geometry, making it an essential tool for problem-solving in mathematics.

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Most popular questions from this chapter

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