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SECTION 1.5 GRAPHING TECHNIQUES; GRAPHING TECHNIQUES; TRANSFORMATIONS TRANSFORMATIONS
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TRANSFORMATIONS Recall our “library” of functions. Here we will learn techniques for graphing a function which is “related” to one we already know how to graph.
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HORIZONTAL SHIFTS On the same screen, graph each of the following functions: Y 1 = x 2 Y 2 = (x - 1) 2 Y 3 = (x - 3) 2 Y 4 = (x + 2) 2
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COMPARING y = x 2 and y = (x - 2) 2 If we named the first function f(x), we could denote the second one by f(x - 2). In general, we can refer to any horizontal shift of a function f(x) by using the notation f(x - h)
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y = f(x - 2)y = f(x + 3) When h is positive (that is, when there is a value being subtracted from x) the shift is to the right. When h is positive (that is, when there is a value being subtracted from x) the shift is to the right. When h is negative (that is, when there is a value being added to x) the shift is to the left. When h is negative (that is, when there is a value being added to x) the shift is to the left.
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VERTICAL SHIFTS In general, we can refer to any vertical shift of a function f(x) by using the notation: f(x) + k
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y = f(x) + 4y = f(x) - 1 When k is positive, the shift is upward. When k is negative, the shift is downward.
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EXAMPLE: The figure shows the graph of f(x). Sketch the graphs of f(x + 1) and f(x) - 1. - 2 - 1 1 2
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y = f(x + 1) - 3 - 2 - 1 1 2
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y = f(x) - 1
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VERTICAL STRETCHES When we compare the graph of y = x 2 to the graph of y = 2x 2, we find the second one is more narrow than the first. When we compare the graph of y = x 2 to the graph of y = 2x 2, we find the second one is more narrow than the first. This is called a vertical stretch. All the y-values are being doubled. This is called a vertical stretch. All the y-values are being doubled.
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VERTICAL SHRINKS When we compare the graph of y = x 2 to the graph of y =.5x 2, we find the second one is wider than the first. When we compare the graph of y = x 2 to the graph of y =.5x 2, we find the second one is wider than the first. This is called a vertical shrink. All the y-values are being halved. This is called a vertical shrink. All the y-values are being halved.
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In general, we can denote vertical stretches and shrinks to a function f(x) in the following way: For a > 1, stretch y = af(x) For 0 < a < 1, shrink
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EXAMPLE: Sketch the graphs of y = 3f(x), y =.5f(x), and y = -.5f(x) Sketch the graphs of y = 3f(x), y =.5f(x), and y = -.5f(x)
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y = 3f (x) - 2 - 1 12
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y =.5f (x)
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y = -.5f (x)
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HORIZONTAL STRETCHES AND SHRINKS In general, we can denote horizontal stretches and shrinks to a function f(x) in the following way: For c > 1, shrink y = f(cx) For 0 < c < 1, stretch
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EXAMPLE: Sketch the graphs of y = f(2x) and y = f(.5x) Sketch the graphs of y = f(2x) and y = f(.5x)
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y = f (2x)
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y = f (.5x)
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EXAMPLE: Given f(x) = x 3 - 4x, explain the transformations that will occur to the graph of the function for f(2x) + 3 The graph will be compressed horizontally and shifted 3 units up. Graph it!
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CONCLUSION OF SECTION 1.5 CONCLUSION OF SECTION 1.5
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