3 4 Graphs and Transformations Define parent functions

3. 4 Graphs and Transformations Define parent functions. Transform graphs of parent functions. 1. 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.

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

Parent Functions greatest integer function quadratic function cubic function

Parent Functions reciprocal function square root function cube root function

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

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

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

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

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

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

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.

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

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 .

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

Combining Transformations 1. If a < 0, reflect over the y-axis. 2. Stretch or compress horizontally by a factor of . 3. Shift the graph horizontally b units left or right. 4. If c < 0, reflect over the x-axis. 5. Stretch or compress vertically by a factor of 6. Shift the graph vertically d units up or down. .

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

Example #6 Combining Transformations Describe the transformations on the following functions, then graph. B. ) 1. 2. 3. 4. 5. 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.