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Lesson 1-3 New Functions from Old Functions Part 1 - Part 1 https://encrypted- tbn3.gstatic.com/images?q=tbn:ANd9GcTMSNbfIIP8t1Gulp87xLpqX92qAZ_vZwe4Q u308QRANh_v4UHWiw

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At this point in time it is important to know and remember certain parent functions and their graphs.

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Now, by recognizing a parent functions graph and then applying certain transformations, we can see how various other graphs can be obtained.

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We are going to want to sketch these other graphs by hand and also come up with their equations.

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Let’s first consider translations. Simply by adding or subtracting a constant c to a formula of a function can cause a slide up, down, right, or left, of a parent functions graph.

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Vertical and Horizontal shifts:

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Vertical and Horizontal shifts: Suppose c > 0, to obtain

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y = f(x) + c shift the graph of y = f(x) a distance of c units upward y = f(x) – c shift the graph of y = f(x) a distance of c units downward y = f(x – c) shift the graph of y = f(x) a distance of c units to the right y = f(x + c) shift the graph of y = f(x) a distance of c units to the left

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Vertical and Horizontal shifts: Suppose c > 0, to obtain y = f(x) + c shift the graph of y = f(x) a distance of c units upward y = f(x) – c shift the graph of y = f(x) a distance of c units downward y = f(x – c) shift the graph of y = f(x) a distance of c units to the right y = f(x + c) shift the graph of y = f(x) a distance of c units to the left

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Vertical and Horizontal shifts: Suppose c > 0, to obtain y = f(x) + c shift the graph of y = f(x) a distance of c units upward y = f(x) – c shift the graph of y = f(x) a distance of c units downward y = f(x – c) shift the graph of y = f(x) a distance of c units to the right y = f(x + c) shift the graph of y = f(x) a distance of c units to the left Catchy little phrase: Add to y means go “high”, add to x means go “west”.

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Example:

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When multiplying or dividing by a constant, this can produce various kinds of stretching, shrinking, or reflections of the graph of a function.

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Vertical and Horizontal Stretching and Reflecting Suppose c > 1, to obtain

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Vertical and Horizontal Stretching and Reflecting Suppose c > 1, to obtain y = cf(x), stretch the graph of y = f(x) vertically by a factor of c y = (1/c)f(x), compress the graph of y = f(x) vertically by a factor of c y = f(cx), compress the graph of y = f(x) horizontally by a factor of c y = f(x/c), stretch the graph of y = f(x) horizontally by a factor of c y = - f(x), reflect the graph of y = f(x) about the x-axis y = f(-x), reflect the graph of y = f(x) about the y-axis

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Example:

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The most common mistake made is to do the transformations in the wrong order!

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Example: The most common mistake made is to do the transformations in the wrong order! (Hint: Follow rules for order of operations.) (pemdas)

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Example:

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Compare the results of y = cf(x) and y = f(cx).

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Now lets take a look at what absolute value | | surrounding a function does to a graph.

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When you take the absolute value of a function f(x), the graph of |f(x)| will be the graph of f(x) except that the part of the graph of f(x) below the x axis will be reflected above the x-axis!

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Take the graph of y = sin x.

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Now, sketch the graph of y = | sin x |.

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Assignment: Pgs. 43-44 #3, 5, 9-23 odd

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