Polynomial Functions that are symmetric about the Y-axis

[flappergirl68]

y= -x^4

is this symmetrical about the y-axis, I didn't know if this fell under that category since the -x is not in paranthesis (wouldn't that mean: (-x)(x)(x)(x)) in which case the positive x reflection of this would not be the same...am I understanding this correctly?

-Thanks!

 

[o_O]

y = -x⁴ is the same thing as saying y = -(x⁴).

Anyway, if it's symmetrical about the y-axis, that would mean it is an even function where: f(x) = f(-x)

f(x) = -x⁴
f(-x) = -[(-x)⁴]

Can you figure out whether or not it's symmetrical about the y-axis now?

 

[flappergirl68]

That would mean that this function doesn't have a reflection about the y-axis, because the answer would always be negative. Am I thinking correctly?

 

[o_O]

Not quite.

Looking at f(x) = -x⁴, it's always negative since anything to an even exponent will be positive and with the negative sign in front, it makes it negative.

Similarly, f(-x) = -(-x)⁴. Will it always be negative or positive?

 

[Deleted member 4993]

Not quite.

Looking at f(x) = -x⁴, it's always negative since anything to an even exponent will be positive and with the negative sign in front, it makes it negative.

Similarly,

f(-x) = -[(-x) ⁴ ] = - [x⁴] = -x⁴ = f(x)

Try putting some real number.

say x = 1

f(1) = -1

f(-1) = -1

say x = 2

f(2) = -16

f(-2) = -16


Now whip out your graphing calculator and plot the function - that should convince you.

Will it always be negative or positive?