Let's start by reviewing the definition of even and odd functions.

A function $y = f (x)$ is even if and only if $f (- x) = f (x)$ for each $x$ in the domain of $f .$ The graph of an even function is symmetric about the $y - axis .$

A function $y = f (x)$ is odd if and only if $f (- x) = - f (x)$ for each $x$ in the domain of $f .$ The graph of an odd function is symmetric about the origin. This means that it looks the same after a reflection in the $x - axis$ followed by a reflection in the $y - axis .$

Recall that a quadratic function of the form

f (x) = a x^{2},

where

a

is any non-zero real number, has its vertex on the origin and is symmetric about the

y - axis .

Therefore, it is an even function. We can verify this by using the definition given above.

f (x) = a x^{2}

Substitute

$x = - x$

f (- x) = a (- x)^{2}

NegBaseToPosPow

$(- a)^{2} = a^{2}$

f (- x) = a x^{2}

Substitute

$a x^{2} = f (x)$

f (- x) = f (x)

Since

f (- x) = f (x)

for each and every

x,

we know it is an even function. This is because squaring the variable eliminates the negative sign. Conversely, this will not happen if the exponent is an odd number. For example, let's consider

g (x) = x^{3} .

g (x) = x^{3}

Substitute

$x = - x$

g (- x) = (- x)^{3}

NegBaseToNegPow

$(- a)^{3} = - a^{3}$

g (- x) = - x^{3}

Substitute

$x^{3} = g (x)$

g (- x) = - g (x)

Since

g (- x) = - g (x)

for each and every

x,

we know that

g (x) = x^{3}

is an odd function. Finally, let's draw both functions together and compare them. For

f (x),

we can choose any non-zero value of

a .

Let's choose

a = \frac{1}{4} .

We will graph the even function

f (x) = \frac{1}{4} x^{2}

and the odd function

g (x) = x^{3} .

Notice that this is just an example, as there are infinitely many odd and even functions.

Exercise