Question 76906: Hi, I have this graph, and it's y-axis goes from -5 to +10. The x-axis goes from +5 to -5. The first section of the graph is a semicircle. It starts a (-3,0) and ends at (3,0). The highest part of the semicircle is at (0,3). Then the second part of the graph is a horizontal line. It starts at (3,0) and ends at (4,4). The slope of that line should be four. Then the third part of the graph should go from (4,4) to (5,0). The slope should be -4. Then the fourth part of the graph should start at (5,0) and end at (6,1). The slope should be 1. The fith part of the graph should be a horizontal line starting at (6,1) and ending at (7,1). Then the final part of the graph starts at (7,1) and ends at (9,0). If you graphed it, it should look like a semicircle, a triangle, and a trapezoid. The x-axis acts as the base in all the shapes. With that graph, I need to find the equation of the line of symmetry for the graph of above each of the following interval if one exists:
[-3,3]________ [3,5]________ [5,9]_________ [-3,9]_________. I have tried to look for the equations, but I have gotten no where. To be honest, I don't even know what the line of symmetry is. Please help me.
Thank You,
Sarah
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Sarah--
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I think I understand what the problem is asking, so I'll take a shot at it. If you have
doubts about what I've done, you will probably need to re-post it so another tutor can have
a shot at it.
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You did a good job of describing the graph. However, in the first line I suspect that you
meant the y-axis goes from -5 to + 5 and the x-axis goes from -5 to +10. You have them reversed.
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This problem involves mostly the ability to visualize things. The lines of symmetry for
the given intervals are the lines about which the figure can be rotated 180 degrees and
still appear to be the same figure. Let's use the semi-circle as our first example.
Suppose you consider that the y-axis is an axle and we can rotate the circle horizontally
around that axle. Maybe you could look at it this way. Take a short piece of wire and
bend it into a semi-circle. Then tape the top center of the semi-circle to a longer
piece of wire that runs vertically through the center of the semi-circle. Then picture
it that you are going to spin the semi-circle around that vertical wire. Every 180 degrees
the semi-circle appears just as it does on your graph. The vertical piece of wire is
the axis of symmetry because the rotating circle is symmetrical ... every 180 degrees
it appears to be the same as it was before you started the rotations. Does that make sense
to you? Maybe it would help to think of an umbrella. If you stood the rod upright on
the floor, you could rotate the umbrella by spinning the rod and to somebody standing away
from you
the cloth semi-circle of the umbrella would always look the same. The semi-circular
cloth part is symmetrical about the handle. Anyhow, the point that I'm trying to make is that
over the interval of x = -3 to x = +3 the semi-circle is symmetrical about the y-axis ...
the side to the left of the y-axis is the same as the side to the right of the y-axis.
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The same can be said of the triangle that exists in the interval x = 3 to x = 5. It is
symmetrical about the vertical line through the point x = 4. In fact, the equation of
this vertical line is x = 4. The left side of this triangle (the side to the left of
the vertical line x = 4 can be rotated about the vertical line, and it will be exactly on
top of the right side of the triangle.
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But what about the trapezoid. You can't find a vertical line of symmetry for that figure
because the slope of the left side of the trapezoid is not the same as the slope of the
right side. Consequently, if you spin the trapezoid about a vertical line, there's
no way that it will look the same every 180 degrees.
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Finally in the span of x = -3 to x = +9 there is no vertical line that you can spin the
entire graph about such that every 180 degrees the figure looks the same. At one point
the trapezoid will be on the right side as it is on your graph, and 180 degrees later it
will be on the left side. That sure is a long way from having a symmetrical figure that
looks the same every 180 degrees.
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So, my take on this problem is:
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[-3,3] the line of symmetry is x = 0 (which is the y-axis).
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[3,5] the line of symmetry is x = 4 (the vertical line through the point x=4]
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[5,9] no line of symmetry
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[-3,9] no line of symmetry
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I hope that this discussion hasn't been too confusing. The type of symmetry we've been
discussing is called "even" symmetry. You need to get a grasp of this concept because
you may be going on to study "odd" symmetry also. And we've been discussing things that
are symmetrical around vertical lines. There are things that are symmetrical about horizontal
lines and also about slanted lines. So it's important for you to get a fundamental
grasp of the concept of symmetry ... visualizing lines about which things can be rotated
without changing the basic figure.
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Hope this helps ... I know I probably have presented it in a confusing manner, but hopefully
you'll be able to grasp the basic concept.
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