1. § 8.3
Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions

2. Graphing Piecewise-Defined Functions
Example:
Graph
Graph each “piece” separately. x
f (x) = 3x – 1
– 1(closed circle) –1
– 4 –2
– 7 x
f (x) = x + 3 1 4 2 5 3 6
Values  0.
Values > 0.
Continued.

3. Graphing Piecewise-Defined Functions
Example continued: x y x
f (x) = 3x – 1
– 1(closed circle) –1
– 4 –2
– 7
(3, 6)
Open circle
(0, 3)
(0, –1) x
f (x) = x + 3 1 4 2 5 3 6
(–1, 4)
(–2, 7)

4. Vertical and Horizontal Shifting
Vertical Shifts (Upward or Downward)
Let k be a Positive Number
Graph of
Same As
Moved
g(x) = f(x) + k
f(x)
k units upward
g(x) = f(x)  k
k units downward
Horizontal Shifts (To the Left or Right)
Let h be a Positive Number
Graph of
Same As
Moved
g(x) = f(x  h)
f(x)
h units to the right
g(x) = f(x + h)
h units to the left

5. Vertical and Horizontal Shifting
Example: x y 5 5
Begin with the graph of f(x) = x2.
Shift the original graph downward 3 units.
(0, –3)

6. Vertical and Horizontal Shifting
Example: x y 5 5
Begin with the graph of f(x) = |x|.
(0, –2)
Shift the original graph to the left 2 units.

7. Reflections of Graphs Reflection about the x-axis
The graph of g(x) = – f(x) is the graph of f(x) reflected about the x-axis.

8. Reflections of Graphs Example:y 5 5
(4, 2)
(4, –2)
Reflect the original graph about the x-axis.