1. 3.4 Graphs and Transformations
Define parent functions.
Transform graphs of parent functions.

2. Parent Functions
Parent functions are used to illustrate the basic shape and characteristics of various functions.
The rules of transforming these functions can be applied to ANY function.

3. Parent Functions constant function identity (linear) function
absolute-value function

4. greatest integer function
Parent Functions
greatest integer function
quadratic function
cubic function

5. Parent Functions reciprocal function square root function
cube root function

6. Vertical Shifts Vertical shift upward c units.
Vertical shift downward c units.

7. Example #1 Shifting a Graph Vertically
Vertical shift 3 units up.
Vertical shift 5 units down.

8. Horizontal Shifts Horizontal shift left c units.
Horizontal shift right c units.

9. Example #2 Shifting a Graph Horizontally
Horizontal shift 2 units right.
Horizontal shift 4 units left.

10. Reflections
Reflection over the x-axis.
Reflection over the y-axis.

11. Example #3 Reflecting a Graph
Reflection over the x-axis.
Reflection over the y-axis.

12. Vertical Stretches & Compressions
Given a function with the transformation:
Every point of the function is changed by
If c > 1, the graph of f is stretched vertically, away from the x-axis, by a factor of c.
If c < 1, the graph of f is compressed vertically, toward the x-axis, by a factor of c.

13. Example #4 Vertical Stretches & Compressions
Vertical compression by a factor of
Vertical stretch by a factor of 2.

14. Horizontal Stretches & Compressions
Given a function with the transformation:
Every point of the function is changed by
If c > 1, the graph of f is compressed horizontally, toward the y-axis, by a factor of
If c < 1, the graph of f is stretched horizontally, away from the y-axis, by a factor of

15. Example #5 Horizontal Stretches & Compressions
Horizontal compression by a factor of
Horizontal stretch by a factor of 5 .

16. Combining Transformations
If a < 0, reflect over the y-axis.
Stretch or compress horizontally by a factor of
Shift the graph horizontally b units left or right.
If c < 0, reflect over the x-axis.
Stretch or compress vertically by a factor of .
Shift the graph vertically d units up or down.

17. Example #6 Combining Transformations
Describe the transformations on the following functions, then graph.
A.)
Horizontal compression by a factor of 1/3.
Shift 4 units right.
Reflect over x-axis.
Shift 2 units up.
Apply transformations using
the order of operations.

18. Example #6 Combining Transformations
Describe the transformations on the following functions, then graph.
B.)
Reflection over the y-axis.
Horizontal stretch by a factor of 4.
Shift 4 units left.
Vertical stretch by a factor of 2.
Shift 4 units down.
Apply transformations using
the order of operations.