Symmetry, Reflections and Inverses - March 14, 2011

Review:

A graph is a function if all the y values have a unique x value.

If it passes the "Vertical Line Test" it means it is a function.

And a function is said to be a one-to-one function if the inverse of the graph is still a function.

Reflection of Functions:

y=f(x) is what you call your "parent graph". Points and equations will be plotted on the Cartesian plane normally.

If a negative one (-1) is added in front of the "parent" equation, so it changes from y=f(x) to y=-f(x), this means that you will have to reflect the graph over the x-axis. This means that the new points of the graph are equal to the points of the "parent" graph when you fold the plane horizontally at the x-axis.

y = f(x) >>>>>>>>>>>>>>>> y = -f(x)

x = 0 3 3 >>>>>>>>>>>>>>>> x = 0 3 -3

y = 2 3 -1 >>>>>>>>>>>>>>>> y= -2 -3 1

If a negative one (-1) is added in front of the x in your parent graph, meaning your equation changed from y=f(x) into y=f(-x), you have to reflect your graph over the y-axis. So this means that your new points would equal you parent graph's points if you were to fold the whole Cartesian plane vertically on the y-axis.

y = f(x) >>>>>>>>>>>>>>>> y = f(-x)

x = 0 3 3 >>>>>>>>>>>>>>>> x = 0 -3 3

y = 2 3 -1 >>>>>>>>>>>>>>>> y= 2 3 -1

Finally if you try to inverse functions that means your "parent" equation changed into y=f^-1(x) you just interchange your x and y values. Your new points should also equal to the parent graph's new points when you fold the Cartesian plane diagonally on y=x .

y = f(x) >>>>>>>>>>>>>>>> y = f^-1(x)

x = 0 3 3 >>>>>>>>>>>>>>>> x = 2 3 -1

y = 2 3 -1 >>>>>>>>>>>>>>>> y= 0 3 3

Finding the Inverse Equation

To find the inverse equation you need to switch the spots of the y and x in the equation.

If the equation is y=4x-7 you change it into x=4y-7 and then you have to rearrange the new equation isolating y. In this case your new equation will be y=(x+7)/3

If there is a specific value for the inverse, such as f^-1(2) , just find the inverse equation first and then substitute the 2 for the x value.

To find f^-1(2), given f(x)=2x+3

f(x)=2x+3

y=2x+3 >>>>>> y=(x+3)/2

f^-1(x)=(x+3)/2 f^-1(x) = y

f^-1(2)=(x+3)/2 >>>>>>> y=(2+3)/2

y=5/2

Therefore your f^-1(2) , in the equation f(x)=2x+3, is ( 2 , 5/2 )

That's it ! The next blogger will be ......... Nick M.