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

The following basic graphs will be used extensively in this section 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 squaring function

The square root function

The absolute value function

The cubing function

The cube root 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 Vertical Translation For b > 0, the graph of y = f(x) + b is the graph of y = f(x) shifted up b units; the graph of y = f(x)  b is the graph of y = f(x) shifted down b units.

Horizontal Translation For d > 0, the graph of y = f(x  d) is the graph of y = f(x) shifted right d units; the graph of y = f(x + d) is the graph of y = f(x) shifted left d units.

Vertical shifts Horizontal 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

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.)

Points represented by (x , y) on the graph of f(x) become If the point (6, -3) is on the graph of f(x), find the corresponding point on the graph of f(x+3) + 2

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.

Reflections The graph of f(x) is the reflection of the graph of f(x) across the x-axis. The graph of f(x) is the reflection of the graph of f(x) across the y-axis. If a point (x, y) is on the graph of f(x), then (x, y) is on the graph of f(x), and (x, y) is on the graph of f(x).

Reflecting Across x-axis (y becomes negative, -f(x)) Across y-axis (x becomes negative, f(-x))

Use the basic graph to sketch the following:

Vertical Stretching and Shrinking The graph of af(x) can be obtained from the graph of f(x) by stretching vertically for |a| > 1, or shrinking vertically for 0 < |a| < 1. For a < 0, the graph is also reflected across the x-axis. (The y-coordinates of the graph of y = af(x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)

VERTICAL STRETCH (SHRINK) y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)

Horizontal Stretching or Shrinking The graph of y = f(cx) can be obtained from the graph of y = f(x) by shrinking horizontally for |c| > 1, or stretching horizontally for 0 < |c| < 1. For c < 0, the graph is also reflected across the y-axis. (The x-coordinates of the graph of y = f(cx) can be obtained by dividing the x-coordinates of the graph of y = f(x) by c.)

Horizontal stretch & shrink We’re MULTIPLYING by an integer (not 1 or 0). x’s do the opposite of what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed horizontally.)

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) 1st 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) 2nd transformation would be 3 times all the y’s, pts. are now (-3,-3), (-2,0), (-1,3) 3rd transformation would be subtract 1 from all y’s, pts. are now (-3,-4), (-2,-1), (-1,2)

Graph of Example (-1,-1), (0,0), (1,1) (-3,-4), (-2,-1), (-1,2)

The point (-12, 4) is on the graph of y = f(x) The point (-12, 4) is on the graph of y = f(x). Find a point on the graph of y = g(x). g(x) = f(x-2) g(x)= 4f(x) g(x) = f(½x) g(x) = -f(x) (-10, 4) (-12, 16) (-24, 4) (-12, -4)