All types of functions have a parent function, the simplest type of function in a family of functions. Some common ones are:
Horizontal
and Vertical shifts: depending on where x is manipulated
will cause a horizontal or vertical shift. If the x is added or subtracted outside
of the parenthesizes, then the graph will shift up or down. If the x is added
or subtracted
inside of the parenthesizes, then the graph will shift left or
right.
Vertical shift up: 
Vertical shift down: 
Horizontal shift to the right: 
Horizontal shift to the left: 
Vertical
Stretch and Horizontal Shrink: a horizontal shrink
and a vertical stretch affect the width of a function. In a vertical stretch or
shrink, the function is multiplied by c outside of the parenthesizes. If c<1, then it
will be a vertical shrink. If c>1, then it will be a vertical stretch. In a
horizontal stretch or shrink, the function is multiplied by c inside of the parenthesizes. If c>1, then it will be a
horizontal shrink. If c<1, then it will be a horizontal stretch.
Vertical stretch: 
Horizontal shrink: 
There are ways to tell if the graph has
been affected vertically or horizontally. In a vertical stretch or shrink,
there is no change in the x-intercepts. In a horizontal stretch of shrink,
there is no change in the relative minimums or maximums or y-intercepts.
Reflections:
a reflection mirrors the function over either the x-axis or y-axis. When –c is multiplied
outside
of the parenthesizes, the function reflects over the x-axis. When –c is
multiplied inside
of the parenthesizes, the function reflects over the y-axis.
Reflection over x-axis: 
Reflection over y-axis: 
Reflections also bring to light the idea of even and odd functions. If f(-x) = f(x), then the function is even. If f(-x)= -f(x), then the function is odd. However, if f(-x) does not equal either, then the function neither odd nor even.
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