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Symmetry and Graphing (page 3 of 3)

Sections: Symmetry about an axis, Symmetry about a point, Symmetry and graphing


Symmetry is more of a geometrical than an algebraic concept, but the subject of symmetry does come up in a couple of algebraic contexts. For instance, when you're graphing quadratics, you may be asked for the parabola's axis of symmetry. This is usually just the vertical line x = h, where
"h" is the x-coordinate of the vertex, (h, k). That is, a parabola's axis of symmetry is usually just the vertical line through its vertex.

The other customary context for symmetry is judging from a graph whether a function is even or odd. Warning: By definition, no function can be symmetric about the x-axis (or any other horizontal line), since anything that is mirrored around an horizontal line will violate the Vertical Line Test. On the other hand, a function can be symmetric about a vertical line or about a point. In particular, a function that is symmetric about the y-axis is also an "even" function, and a function that is symmetric about the origin is also an "odd" function. Because of this correspondence between the symmetry of the graph and the evenness or oddness of the function, "symmetry" in algebra is usually going to apply to the y-axis and the origin.

  • For each of the following graphs, list any symmetries, and state whether the graph shows a function.
       
  • A: the graph of a parabola

    B: the graph of an hyperbola

    C: the graph of an ellipse

    D: the graph of an hyperbola

     

    E: the graph of a square-root function

    F: the graph of a cubic function

    G: the graph of a parabola

    H: the graph of a parabola

    Graph A: This graph is symmetric about its axis, the line x = 3. There is no other symmetry. This graph shows a function. Copyright © Elizabeth Stapel 2005-2011 All Rights Reserved

    Graph B: This graph is symmetric about the axes x = 0 (the y-axis) and y = 0 (the x-axis), and also about the origin. Since there exist vertical lines (such as the line x = 2) which will cross this graph twice, it does not show a function.

    Graph C: This graph is symmetric about the axes x = 1 and y = –2, and symmetric about the point (1, –2). Since a vertical line can be drawn to cross the ellipse twice, this is not a function.

     

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    Graph D: This graph is symmetric about slanty lines: y = x and y = –x. It is also symmetric about the origin. Because this hyperbola is angled correctly (so that no vertical line can cross the graph more than once), the graph shows a function.

    Graph E: This graph (of a square-root function) shows no symmetry whatsoever, but it is a function.

    Graph F: This graph (of a cubic function) is symmetric about the point (–4, –1), but not around any lines. This graph does show a function.

    Graph G: This parabola is lying on its side. It is symmetric about the line y = 2. It is not a function.

    Graph H: This parabola is vertical, and is symmetric about the y-axis. It is a function; in fact, it is an even function.


As mentioned earlier, you'll mostly only be looking for symmetry about the y-axis (that is, for whether the function is even) or about the origin (that is, for whether the function is odd).

  • Determine from the graphs whether the displayed functions are even, odd, or neither.
      
  • A: the graph of a straight line

    B: the graph of a negative quadratic

    C: the graph of a cubic

    D: the graph of a cubic

     

    E: the graph of a cube root

    F: the graph of a square root

    G: a graph that looks something like a bell-shaped curve

    H: the graph of an hyperbola

    Graph A: This linear graph goes through the origin. If I rotate the graph 180° around the origin, I'll get the same picture. So this graph is odd. (The function would not be odd if this line didn't go through the origin.)

    Graph B: This parabola's vertex is on the y-axis, so the axis of symmetry is the y-axis. That means that the function is even.

    Graph C: This cubic is centered on the origin. If I rotate the graph 180° around the origin, I'll get the same picture. So this graph is odd.

    Graph D: This cubic is centered at the point (0, –3). This graph is symmetric, but not about the origin or the y-axis. So this function is neither even nor odd.

    Graph E: This cube root is centered on the origin, so this function is odd.

    Graph F: This square root has no symmetry. The function is neither even nor odd.

    Graph G: This graph looks like a bell-shaped curve. Since it is mirrored around the y-axis, the function is even.

    Graph H: This hyperbola is symmetric about the lines y = x and y = –x, but this tells me nothing about evenness or oddness. But the graph is also symmetric about the origin, so this function is odd.

When looking for symmetry, you don't have to just sit there thinking the thing through. Instead, take the paper and your pencil, and see if there is a spot where you can plant the pencil's eraser, and then spin the paper on the table. When it's spun halfway around, do you get the same picture as you had before? Then your eraser marks a point of symmetry. Grab a ruler and stand it on its edge in the middle of the graph. Look down onto the paper, and eye-ball the two "sides" of the picture. Do the two portions of the graph, one on either side of the ruler, look like mirror images? Then the ruler marks a line of symmetry. Don't be shy about putting your hands into the work; it can really help in getting a "feel" for symmetry.

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Cite this article as:

Stapel, Elizabeth. "Symmetry and Graphing." Purplemath. Available from
    http://www.purplemath.com/modules/symmetry3.htm. Accessed
 

 

 

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