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**3.4 Graphs and Transformations**

Define parent functions. Transform graphs of parent functions.

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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.

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**Parent Functions constant function identity (linear) function**

absolute-value function

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**greatest integer function**

Parent Functions greatest integer function quadratic function cubic function

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**Parent Functions reciprocal function square root function**

cube root function

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**Vertical Shifts Vertical shift upward c units.**

Vertical shift downward c units.

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**Example #1 Shifting a Graph Vertically**

Vertical shift 3 units up. Vertical shift 5 units down.

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**Horizontal Shifts Horizontal shift left c units.**

Horizontal shift right c units.

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**Example #2 Shifting a Graph Horizontally**

Horizontal shift 2 units right. Horizontal shift 4 units left.

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

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**Example #3 Reflecting a Graph**

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

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**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.

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**Example #4 Vertical Stretches & Compressions**

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

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**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

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**Example #5 Horizontal Stretches & Compressions**

Horizontal compression by a factor of Horizontal stretch by a factor of 5 .

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**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.

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**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.

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**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.

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