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Reference

Symmetry in Functions

Concept

Axis of Symmetry

An axis of symmetry is the line that divides the graph of a function into two mirrored images. Consider the graph of a quadratic function which is a parabola that has an axis of symmetry that is parallel to the

y -

axis and passes through the vertex.

It is also possible for a function to have an axis of symmetry which is not vertical. Take the following graph as an example.

The graph of the example function has an inclined axis of symmetry

What is more, if a function's axis of symmetry is the $y -$ axis, it is an even function.

Function 0.8*x^4-3*x^2+1.5 is symmetric with respect to the y-axis.

The concept of an axis of symmetry extends beyond graphs of functions. Any geometric figure can have an axis of symmetry if a line exists that divides the figure into congruent, mirror-image halves.

Concept

Even Function

An even function is a function for which the value of $f (- x)$ is equal to the value of $f (x)$ for all the values in its domain. That is, opposite inputs have the same output.

$f (- x) = f (x)$

The graph of an even function is symmetric about the

y -

axis. The functions

y = x^{2}

and

y = 2 ∣ x ∣

are two examples of even functions.

Graphs of f(x)=x^2 and graph of g(x)=2|x|

Notice that if a function is even and the point

(x, y)

is on the graph, then the point

(- x, y)

is also on the graph. A function can be even, odd, or neither.

Extra

Determining If a Function is Even

To determine algebraically whether a function is even, substitute

- x

into the function rule and simplify. If the resulting expression is equal to

f (x),

then the function is even; otherwise, it is not. For example, consider the following function.

f (x) = 3 x^{4} - 2 x^{2} + 1

Substitute

- x

for

x

and simplify.

f (x) = 3 x^{4} - 2 x^{2} + 1

Substitute

$x = - x$

f (- x) = 3 (- x)^{4} - 2 (- x)^{2} + 1

NegBaseToPosPow

$(- a)^{4} = a^{4}$

f (- x) = 3 x^{4} - 2 (- x)^{2} + 1

NegBaseToPosBase

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

f (- x) = 3 x^{4} - 2 x^{2} + 1

Substitute

$3 x^{4} - 2 x^{2} + 1 = f (x)$

f (- x) = f (x)

Since

f (- x) = f (x),

the given function is even.

Concept

Odd Function

An odd function is a function for which the value of $f (- x)$ is equal to the value of $- f (x)$ for all the values in its domain. It is like if the function allows moving the negative sign from the input to the output.

$f (- x) = - f (x)$

The graph of an odd function is symmetric about the origin, meaning that the graph looks the same after a

18 0^{\circ}

rotation about the origin. The functions

y = x

and

y = x^{3}

are two examples of odd functions.

Notice that if a function is odd and the point

(x, y)

is on the graph, then the point

(- x, - y)

is also on the graph. A function can be odd, even, or neither.

Extra

Determining If a Function is Odd

To determine algebraically whether a function is odd, substitute

- x

into the function rule and simplify. If the resulting expression is equal to

- f (x),

then the function is odd; otherwise, it is not. For example, consider the following function.

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

Substitute

- x

for

x

and simplify.

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

Substitute

$x = - x$

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

NegBaseToPosBase

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

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

MoveNegNumToFrac

Put minus sign in front of fraction

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

Substitute

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

f (- x) = - f (x)

Since

f (- x) = - f (x),

the given function is odd.

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