Today, we learned about Symmetry, Reflections and Inverses!
When x is replaced with its reciprocal (-x) in the equation of a function y= f(x), it's graph is expressed in the y-axis.
For example,
f(x)= x³
f(-x) = (-x³)
The reflection on the y-axis --> make the 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.
For example:
f (x)= x²
-f (x) = (x²) --> f(x)= -(x²)
Reflection on 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.
The process of finding the Inverse Function:
1) Replace f (x) with y.
2) Switch x and y.
3) Solve for y.
4) Replace with f -¹ (x)
Graph the function f (x)= 2x+2 and its inverse. Determine algebraically the equation of the f -¹ (x).
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| Reflection in the mirror line --> switch x and y values! |
1) y= 2x+2
Remember: y= mx+ b
m= rise/ run
b= y-intercept
(1,4) -> (4,1)
(0,2) -> ( 2,0)
(1,0) -> (0,1)
(-2,-2) -> (-2,-2)
3)
f(x)= 2x+2
y= 2x+2
x= 2y +2
x-2/2= 2y/2
y= x-2/2
y= 1/2 x-1
f -¹ (x) = 1/2 x-1
Transformations
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Effects on Graph
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-f (x)
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Reflection in x-axis
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f (-x)
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Reflection in y-axis
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f-¹ (x)
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Reflection in y=x
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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)
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)
Determining if a shape is a function
When is a relation a function?
A relation is a function if each x values has 1 y values.
There are two tests to check:
VLT --> Vertical line test
HLT--> Horizontal line test
Use the VLT test first if it passes and then use HLT.
VLT- if a vertical line test crosses through the graph only once then the graph is a function, IF more than once it's not a function
HLT- if a horizontal line crosses through the graph only once and it has already passed the VLT then the graph is a one-to-one function.
Not a function:
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| It touched the line twice while performing a vertical line test |
Don't forget to do our homework!
Mr. P gave us an Inverse Function sheet and Exercise 9, Questions 1-20, Omit 5a iii & iv, 5b, 10, 16, and 17 
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