Transformation of Functions Recognize graphs of common functions Use reflections to graph functions Use shifts to graph functions Use narrows and widens to transform graphs Graph functions with sequence of transformations

Function family: Equation:

We will now see how certain transformations (operations) of a function change its graph. This will give us a better idea of how to quickly sketch the graph of certain functions. The transformations are (1) translations, (2) reflections, and (3) narrows or widens.

Horizontal (Left or Right) Translation The graph of y = a(x-h) 2 + k If inside the parenthesis is (x + a number) then the parent graph shifts left h units. If inside the parenthesis is (x - a number) then the parent graph shifts right h units.

Vertical (Up or Down) Translation The graph of y = a(x-h) 2 + k If k> 0 then the parent graph is shifted up k units. If k< 0 then the parent graph is shifted down k units.

Vertical shifts –Moves the graph up or down –Impacts only the “y” values of the function –No changes are made to the “x” values Horizontal shifts –Moves the graph left or right –Impacts only the “x” values of the function –No changes are made to the “y” values

Recognizing the shift from the equation, examples of shifting the function f(x) = Vertical shift of 3 units up? Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)?

Explain how each is different from the parent graph.

Combining a vertical & horizontal shift Example of function that is shifted down 4 units and right 6 units from the original function. If is the parent function, what is the equation of the function shown in red?

Explain how the graphs change from the parent graphs in the following.

Narrows or Widens The graph of f(x)=a(x-h) 2 + k transforms the parent graph: narrows for |a| > 1, or widens for 0 < |a| < 1. For a < 0, the graph is also reflected across the x-axis.

Sequence of transformations Follow order of operations. Select two points (or more) from the original function and move that point one step at a time. f(x) contains (-1,-1), (0,0), (1,1) 1 st transformation would be (x+2), which moves the function left 2 units (subtract 2 from each x), pts. are now (-3,-1), (-2,0), (-1,1) 2 nd transformation would be 3 times all the y’s, pts. are now (-3,-3), (-2,0), (-1,3) 3 rd transformation would be subtract 1 from all y’s, pts. are now (-3,-4), (-2,-1), (-1,2)