# Polynomial Functions that are symmetric about the Y-axis

#### flappergirl68

##### New member
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

##### Full Member
y = -x<sup>4</sup> is the same thing as saying y = -(x<sup>4</sup>).

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

f(x) = -x<sup>4</sup>
f(-x) = -[(-x)<sup>4</sup>]

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

#### flappergirl68

##### New member
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

##### Full Member
Not quite.

Looking at f(x) = -x<sup>4</sup>, 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)<sup>4</sup>. Will it always be negative or positive?

#### Subhotosh Khan

##### Super Moderator
Staff member
o_O said:
Not quite.

Looking at f(x) = -x<sup>4</sup>, 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) <sup>4</sup> ] = - [x<sup>4</sup>] = -x<sup>4</sup> = 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?