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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.

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

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) = 3x^4 - 2x^2 + 1 Substitute -x for x and simplify.

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

Substitute

x= -x

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

NegBaseToPosPow

(- a)^4 = a^4

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

NegBaseToPosBase

(- a)^2=a^2

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

Substitute

3x^4 - 2x^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 180^(∘) 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) = x/x^2-1 Substitute -x for x and simplify.

f(x) = x/x^2-1

Substitute

x= -x

f( -x) = -x/( -x)^2-1

NegBaseToPosBase

(- a)^2=a^2

f(-x) = -x/x^2-1

MoveNegNumToFrac

Put minus sign in front of fraction

f(-x) = -x/x^2-1

Substitute

x/x^2-1= f(x)