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A library of Elementary Graphs
Shifting Graphs Horizontally and Vertically
Reflecting Graphs
Stretching and Shrinking Graphs
Even and Odd Functions
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Transformation – The graph of a new function that is formed by
performing an operation on a given function.
 E.X. – Add constant K to f(x) the graph of y = f(x) is transformed
to y = f(x) + k
Shift - A shift is a rigid translation in that it does not change the shape
or size of the graph of the function. All that a shift will do is change
the location of the graph.
Vertical Shifting - A vertical shift adds/subtracts a constant to/from
every y-coordinate while leaving the x-coordinate unchanged.
Horizontal Shifting - A horizontal shift adds/subtracts a constant
to/from every x-coordinate while leaving the y-coordinate
unchanged. Vertical and horizontal shifts can be combined into one
expression.
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Review 2-1 for reflections of graphs and
symmetry properties
NOW – Consider reflection as an operation
that transforms the graph of a function
 Reflection through the x axis (changing your y-
coordinate sign)
 Reflection through the y axis (changing your xcoordinate sign)
 Reflection through the origin (changing the sign on
both coordinates)
Stretching the graph of f vertically moves you away from the xaxis
Shrinking the graph of f vertically moves you toward the x- axis
Shrinking the graph of f horizontally moves you toward the y- axis
Stretching the graph of f horizontally moves you away from the yaxis
Considered non-rigid transformations because they change the
shape of the graph by either stretching or shrinking it.
Vertical Shift
y = f(x) + k
{k >0 Shift graph of y = f(x) up K units
{k <0 Shift graph of y = f(x) down K units
Horizontal Shift
y = f(x+h)
{h > 0 Shift graph of y = f (x) left h units
{h < 0 Shift graph of y = f(x) right  h units
Vertical Stretch and Shrink
y = Af(X)
{ A > 1 Vertically stretch the graph of y = f(x)
by multiplying each y value by A
{ 0 < A <1 Vertically shrink the graph of y = f(x)
by multiplying each y value by A
Horizontal Stretch and Shrink
y = f(Ax)
{ A> 1 Horizontally shrink the graph of y = f(x)
by multiplying each x value by 1/A
{ 0 < A < 1 Horizontally stretch the graph of y = f(x)
by multiplying each x value by 1/A
Reflection
y = -f(x)
y = f(-x)
y = -f(-x)
Reflect the graph of y=f(x) through the x axis
Reflect the graph of y=f(x) through the y axis
Reflect the graph of y=f(x) through the origin
Certain transformations leave the graph of some
functions unchanged.
If f(x) = f(-x) for all x in the domain of f, then f is an
even function and is symmetric with respect to the y
axis
E.X. reflecting the graph of x2
If f(-x) = -f(x) for all x in the domain of f, then f is an
odd function and is symmetric with respect to the
origin
E.X. reflecting the graph of x3