2.6 transformations of functions Parent Functions and Transformations

Transformation of Functions Recognize graphs of common functions Use shifts to graph functions Use reflections to graph functions Graph functions w/ sequence of transformations

The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.

The identity function f(x) = x

The quadratic function

The square root function

The absolute value function

The cubic function

The rational function

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) stretching.

Vertical Translation OUTSIDE IS TRUE! Vertical Translation the graph of y = f(x) + d is the graph of y = f(x) shifted up d units; the graph of y = f(x)  d is the graph of y = f(x) shifted down d units.

Horizontal Translation INSIDE LIES! Horizontal Translation the graph of y = f(x  c) is the graph of y = f(x) shifted right c units; the graph of y = f(x + c) is the graph of y = f(x) shifted left c units.

The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function. Numbers added or subtracted inside translate left or right, while numbers added or subtracted outside translate up or down.

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.)

Use the basic graph to sketch the following:

Combining a vertical & horizontal shift Example of function that is shifted down 4 units and right 6 units from the original function.

Use the basic graph to sketch the following:

The big picture…

Example Write the equation of the graph obtained when the parent graph is translated 4 units left and 7 units down.

Example Explain the difference in the graphs Horizontal Shift Left 3 Units Vertical Shift Up 3 Units

Describe the differences between the graphs Try graphing them…

A combination If the parent function is Describe the graph of The parent would be horizontally shifted right 3 units and vertically shifted up 6 units

If the parent function is What do we know about The graph would be vertically shifted down 5 units and vertically stretched two times as much.

What can we tell about this graph? It would be a cubic function reflected across the x-axis and horizontally compressed by a factor of ½.