Question

What is the difference between odd functions and even functions?

Answer 1

@ranga

Answer 2

"Even functions are symmetrical, while odd functions are not. If a function is raised to an odd power, it will be an odd function, and if it is raised to and even power, it will be an even function. Therefore, x^3 is asymmetrical, and x^4 is symmetrical."

That is what I have right now as my answer, is it close to what it should be?

Answer 3

It is pretty good so far except that x^3 is also symmetric -- but it is symmetric about the origin. x^4 is symmetric with respect to the y-axis.

In an even function, if you replace x with -x, you will get the same function. Therefore, the y-value will be the same whether x is, say +3 or -3 and therefore even functions are symmetric about the y-axis.

Answer 4

y = x^4
replace x with -x:
y = (-x)^4 = x^4 (same function as before)
so x = -1 and x = +1 will give the same y-value
x = -2 and x = +2 will give the same y-value.
Therefore, the function is symmetric about the y-axis.

y = x^3 is an odd function because replacing x with -x changes the function:
y = (-x)^3 = -x^3 which is different from y = x^3.
Since replacing x with -x changes the sign of the function it is an odd function.

There are also functions that neither even nor odd.
y = x^2 + 2x
replace x with -x
y = (-x)^2 + (2)(-x) = x^2 - 2x which is neither the original function nor the negative of the original function and so this is nether even nor odd function.

Answer 5

translating means moving the function around.
If I plot the function: y = x^2 it will be a parabola with the origin as the vertex.
If I plot y = x^2 + 3, this will also be a parabola but shifted UP along the y-axis by 3 units. This "shifting" is called translating.

y = (x - 5)^2 will shift the graph y = x^2 by 5 units to the RIGHT.