A library of Elementary Graphs Shifting Graphs ...riesg036/doc/transformations.pdf · A library of Elementary Graphs Shifting Graphs Horizontally and Vertically Reflecting Graphs

A library of Elementary Graphs Shifting Graphs Horizontally and Vertically Reflecting Graphs Stretching and Shrinking Graphs Even and Odd Functions

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 transformedto 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.

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 x-coordinate sign)

Reflection through the origin (changing the sign on both coordinates)

Stretching the graph of f vertically moves you away from the x-axis

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 y-axis

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) Reflect the graph of y=f(x) through the x axisy = f(-x) Reflect the graph of y=f(x) through the y axisy = -f(-x) 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