SOLUTION: Determine the symmetry (relative to the x-axis, y-axis, and/or orgin) of the graph of the following function. No need to submit graph. g(x) = 3x^2.

Algebra ->  Graphs -> SOLUTION: Determine the symmetry (relative to the x-axis, y-axis, and/or orgin) of the graph of the following function. No need to submit graph. g(x) = 3x^2.       Log On


   

Question 87303: Determine the symmetry (relative to the x-axis, y-axis, and/or orgin) of the graph of the following function. No need to submit graph. g(x) = 3x^2.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Remember, a function is even when this equality is true:

f%28x%29=f%28-x%29

and a function is odd when this equality is true:

-f%28x%29=f%28-x%29

Lets see if the function is even:

y=3x%5E2 Start with the given equation
Here's f%28x%29:
f%28x%29=3x%5E2 This is the given equation
Here's f%28-x%29:
f%28-x%29=3%28-x%29%5E2 Replace x with -x

f%28-x%29=3x%5E2 Simplify
Since 3x%5E2=3x%5E2, this means the equation f%28x%29=f%28-x%29 is true.

Since f%28x%29=f%28-x%29, the equation y=3x%5E2 is an even function. This means the equation has symmetry with respect to the y axis



Now lets see if the function is odd:

y=3x%5E2 Start with the given equation
Here's f%28-x%29:
f%28-x%29=3x%5E2 Remember we solved for this previously
Here's -f%28x%29:
-f%28x%29=-%283x%5E2%29 Negate the whole function

f%28-x%29=-3x%5E2 Distribute the negative and simplify
Since -3x%5E2 does not equal 3x%5E2, this means the equation -f%28x%29=f%28-x%29 is not true.

Since -f%28x%29 does not equal f%28-x%29, the equation y=3x%5E2 is not an odd function.

When we graph the equation y=3x%5E2, we can see if the equation has any symmetry:
+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+3x%5E2%29+
and we can clearly see that the equation is symmetrical with respect to the y axis
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