SOLUTION: without graphing, determine whether each equation has a graph that is symmetric with respect to the x-axis, the origin, or none of these. a. Y=2x4-3(Note:the x is to the 4th po

Algebra ->  Graphs -> SOLUTION: without graphing, determine whether each equation has a graph that is symmetric with respect to the x-axis, the origin, or none of these. a. Y=2x4-3(Note:the x is to the 4th po      Log On


   



Question 979372: without graphing, determine whether each equation has a graph that is symmetric with respect to the x-axis, the origin, or none of these.
a. Y=2x4-3(Note:the x is to the 4th power, I don't know how to write it on here.)
b. y=x+15

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

a. y=2x%5E4-3
We can often see symmetry visually, but to be really sure we should check a simple fact:
For symmetry with respect to the y-axis, check to see if the equation is the same when we replace x with -x:

is y=2x%5E4-3 symmetric about the y-axis?
Try to replace x with -x:
y=2%28-x%29%5E4-3
Since %28-x%29%5E4+=+x%5E4 (multiplying a negative times a negative gives a positive), there is no change.
So, y=2x%5E4-3+ is symmetric about the y-axis

For symmetry about x-axis use the same idea as for the y-axis, but try replacing y with -y.
-y=2x%5E4-3+
-y%2F-1=2x%5E4%2F-1-3%2F-1+
y=-2x%5E4%2B3+
Now try to get the original equation:
2x%5E4-3%3C%3E-2x%5E4%2B3+ => It is different.
Hence y=2x%5E4-3+ is not symmetric about the x-axis

Diagonal symmetry:
Try swapping y and x :
x=2y%5E4-3+
rearrange :
x%2B3=2y%5E4+
x%2F2%2B3%2F2=y%5E4+
y=root%284%2Cx%2F2%2B3%2F2%29+
And we do not have the original equation. They are not the same.
Hence y=2x%5E4-3+ has not diagonal symmetry.


Origin symmetry is when every part has a matching part:
the same distance from the central point
but in the opposite direction.
Check to see if the equation is the same when we replace both x with -x and y with -y.
does y=2x%5E4-3+ have origin symmetry?
Start with:
y=2x%5E4-3+
Replace x with -x and y with -y:
-y=2%28-x%29%5E4-3+
-y=2x%5E4-3+
the left side is different from the original equation and the right side is identical to the original equation, and this isn’t equivalent to the original equation and we do not have symmetry about the origin

so, y=2x%5E4-3+ is not symmetric about the origin

b.
y=x%2B15
to check is it symmetric about the y-axis, replace x with -x:
y=-x%2B15=> is different from the original equation, so it is not symmetric about the y-axis
to check is it symmetric about the x-axis, replace y with -y:
-y=x%2B15=> is different from the original equation, so it is not symmetric about the x-axis

To test for symmetry over the origin, you replace x with -x, and y+with+-y.
Again if the equation is identical to the original equation, then the graph is symmetric over the origin.
-y=-x%2B15=> the equation is not identical to the original equation, so it is not symmetric over the origin
remember: The linear equations y+=+x or +y+=+-x are the only two equations which are symmetric over the origin.
Note that most graphs don’t have any kind of symmetry. Also, it is possible for a graph to have more than one kind of symmetry. For example the graph of a circle centered at the origin exhibits all three symmetries.