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) (Show Source):
You can put this solution on YOUR website!
a.
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 with :
is symmetric about the y-axis?
Try to replace with :
Since (multiplying a negative times a negative gives a positive), there is change.
So, symmetric about the y-axis
For symmetry about x-axis use the same idea as for the y-axis, but try replacing with .

Now try to get the original equation:
=> It is .
Hence is symmetric about the x-axis
Diagonal symmetry:
Try swapping and :
rearrange :
And we do not have the original equation. They are not the same.
Hence has 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 with and with .
does have origin symmetry?
Start with:
Replace with and with :

the left side is 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 have symmetry about the origin
so, is symmetric about the origin
b.

to check is it symmetric about the y-axis, replace with :
=> 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 with :
=> is different from the original equation, so it is not symmetric about the x-axis
To test for symmetry over the origin, you replace with , and with .
Again if the equation is identical to the original equation, then the graph is symmetric over the origin.
=> the equation is not identical to the original equation, so it is not symmetric over the origin
remember: The linear equations or 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.
|
|
|