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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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