3.4 Graphs and Transformations

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Presentation transcript:

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

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

greatest integer 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 compression by a factor of . Vertical stretch by a factor of 2.

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 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.

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.

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.