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Parent Functions and Transformations

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Transformation of Functions Recognize graphs of common functions Use shifts to graph functions Use reflections to graph functions Graph functions w/ sequence of transformations

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The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.

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The identity function f(x) = x

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The quadratic function

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The square root function

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The absolute value function

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The cubic function

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The rational function

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We will now see how certain transformations (operations) of a function change its graph. This will give us a better idea of how to quickly sketch the graph of certain functions. The transformations are (1) translations, (2) reflections, and (3) stretching.

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Vertical Translation OUTSIDE IS TRUE! Vertical Translation the graph of y = f(x) + d is the graph of y = f(x) shifted up d units; the graph of y = f(x) d is the graph of y = f(x) shifted down d units.

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Horizontal Translation INSIDE LIES! Horizontal Translation the graph of y = f(x c) is the graph of y = f(x) shifted right c units; the graph of y = f(x + c) is the graph of y = f(x) shifted left c units.

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The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function. Numbers added or subtracted inside translate left or right, while numbers added or subtracted outside translate up or down.

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Recognizing the shift from the equation, examples of shifting the function f(x) = Vertical shift of 3 units up Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

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Use the basic graph to sketch the following:

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Combining a vertical & horizontal shift Example of function that is shifted down 4 units and right 6 units from the original function.

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Use the basic graph to sketch the following:

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The big picture…

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Example Write the equation of the graph obtained when the parent graph is translated 4 units left and 7 units down.

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Example Explain the difference in the graphs Horizontal Shift Left 3 Units Vertical Shift Up 3 Units

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Describe the differences between the graphs Try graphing them…

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A combination If the parent function is Describe the graph of The parent would be horizontally shifted right 3 units and vertically shifted up 6 units

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If the parent function is What do we know about The graph would be vertically shifted down 5 units and vertically stretched two times as much.

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What can we tell about this graph? It would be a cubic function reflected across the x-axis and horizontally compressed by a factor of ½.

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