Section 1.3 Shifting, Reflecting, and Stretching Graphs

In this section, you will learn how to: Shift graphs (left, right, up, down) Reflect graphs (across the x or y axis) Stretch graphs (or shrink) You may also look at a graph and be asked to explain what happened to the “original” graph. You must first recognize the graphs of common functions.

p. 100 Common Functions

Vertical and Horizontal Shifts Let c be a positive real number. Vertical and horizontal shifts in the graph of y=f(x) are represented as follows. 1. Vertical shift c units upward : h(x)= f(x) + c 2. Vertical shift c units down : h(x)= f(x) - c 3. Horizontal shift c units right : h(x)= f(x - c) 4. Horizontal shift c units left : h(x)= f(x + c) ***Notice on the horizontal shift, you move OPPOSITE of what you think you would. If the shift happens outside the function, it is vertical. If the shift happens inside the function, it is horizontal.

Example 1 Compare the graph of each function with the graph of Shifts down 1. Shifts to the right 1. Shifts to the left 2 and up 1.

The graphs shown are shifts of the graph of Find equations for g and h..)( 2 xxf  y=g(x) y=h(x) y=f(x)

Reflections

h( x )= - f(x) reflects across the x axis. h( x )= f(-x) reflects across the y axis. If it happens outside the function, reflects across the x axis. If it happens inside the function, reflects across the y axis.

y=g(x) y=h(x)

Example 4 p. 104 Compare the graph of each function with the graph of Reflects across the x axis. Reflects across the y axis. Reflects across the x axis. Shifts to the left 2 units.

Transformations Rigid transformations don’t change the basic shape of the graph. Nonrigid transformations do change or distort the basic shape of the graph. The graph of y = f (x) is represented by y = cf (x). “ c ” represents a constant #. Vertical Stretch if the function is multiplied by a number >1. Vertical Shrink if the function is multiplied by a fraction between 0 and 1.

Example h(x) = 3|x|Vertical stretch

Example Vertical shrink

Specify the sequence of transformations that will yield the graph of the given function from the graph of the function This graph is going down 10 units, followed by a vertical stretch. This graph is going down 2 units followed by a vertical stretch.

“Nobody ever uses this!!!” Tilt-Shift Photography

Tilt Shift Video The mathematics…