Symmetry, Reflections and Inverses

Today, we talked about the symmetry, reflections and inverses.
When x is replaced with -x in the equation of a function y= f(x), its graph is reflected in the y-axis.

f(x) = x³
f(-x) = (-x³)












Reflection in the y-axis -> make x-values negative!

When y is replaced with -y in the equation of a function y= f(x), its graph is reflected in the x-axis.
f(x) = x²
-f(x) = (x²) -> f(x)= -(x²)



















Reflection in the x-axis -> make y-values negative!
When x is interchanged with y in the equation of a function y= f(x), it's reflected in the mirror line y=x. This is called an inverse function.

Finding the Inverse Equation Steps
Replace f(x) with y
Switch x and y
Solve for y
Replace y with f^-1 (x)

Graph the function f(x)=2x+2 and its inverse. Determine algebraically the equation of the f^-1(x).





(1,4) -> (4,1)

(0,2) -> ( 2,0)

(1,0) -> (0,1)

(-2,-2) -> (-2,-2)






f(x)=2x+2
y=2x+2
x=2y+2
2y/2=x-2/2
y=x-2/2

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

Reflection in the mirror line ->switch x and y values!

Generalizing Reflections
Transformations Effect on Graph
-f(x) Reflection in x-axis
f(-x) Reflection in y-axis
f^-1 Reflection in y=x

Symmetry
A graph is said to be symmetrical through an axis or the origin if either side is the mirror image of the other.

A function f(x) is even if for any value "x" f(-x) or -f(-x)=-f(x). Even functions are symmetric about the y-axis. This means that positive and negative x-values result in the same y-value. Even functions would be symmetrical between quadrants 1 and 2 or quadrants 3 and 4. (example is a vertical parabola)

A function f(x) is odd if a any value "x" f(-x) = -f(x) or f(x)=-f(-x). Odd functions are symmetric about the origin. This means that positive and negative x-values result in different y-values. Odd functions would be symmetrical between quadrants 1 and 3 or quadrants 2 and 4 (example is a vertical cube)