Functions+Notes+_updated_.pdf

Solution the graph of y x 2 4 x 3 is a concave up

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Solution: The graph of y = x 2 - 4 x + 3 is a concave up parabola. Completing the square we have that y = x 2 - 4 x + 3 = x 2 - 4 x + 4 - 1 = ( x - 2) 2 - 1 So the graph of y = x 2 - 4 x + 3, or y = ( x - 2) 2 - 1 is the graph of y = x 2 shifted to the right 2 units and down 1 unit. -1 2 0 -1 x y y = x 2 y = ( x - 2) 2 - 1 y = ( x - 2) 2 Example 2.22. Sketch the graph of the rational function f ( x ) = 2 - x x - 1 shifting the graph of y = 1 /x . Solution: It is not immediately obvious that this graph is a shifted version of the graph of y = 1 /x . To see that it is, we can divide x - 1 into 2 - x to get a quotient of - 1 and a remainder of 1: 2 - x x - 1 = - x + 1 + 1 x - 1 = - ( x - 1) + 1 x - 1 = - 1 + 1 x - 1 . Thus, the graph is that of 1 /x shifted to the right 1 unit and down 1 unit.
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2. Functions 33 Example 2.23. Sketch the graph of the function f ( x ) = 1 + | x - 2 | shifting the graph of y = | x | . Express f as a piecewise-defined function. Solution: The graph of f is the graph of the absolute value function shifted to the right 2 units and up 1 unit. 2 0 1 x y y = | x | y = | x - 2 | y = 1 + | x - 2 | Keeping in mind that | x - 2 | = x - 2 if x 2, and | x - 2 | = - ( x - 2) = - x +2 if x < 2, we have that f ( x ) = 1 + ( - x + 2) if x < 2 1 + ( x - 2) if x 2 = - x + 3 if x < 2 x - 1 if x 2
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34 J. S´ anchez-Ortega Stretching and Reflecting transformations Let c be a real number with c > 1. Then the graph of y = cf ( x ) is the graph of y = f ( x ) stretched by a factor of c in the vertical direction (because each y -coordinate is multiplied by the same number c ). The graph of y = - f ( x ) is the graph of y = f ( x ) reflected about the x -axis because the point ( x, y ) is replaced by the point ( x, - y ). See the table below for more stretching, shrinking and reflections transformations.
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2. Functions 35 Example 2.24. In the next figure, we could see some transformations of the cosine function. For example, to get the graph of y = 2 cos x we multiply the y -coordinate of each point on the graph of y = cos x by 2. This means that the graph of y = cos x gets stretched vertically by a factor of 2. Example 2.25. To obtain the graph of the concave down parabola y = - x 2 , we have just to reflect the graph of the concave up parabola y = x 2 about the x -axis. -1 1 0 x y y = x 2 y = - x 2 Example 2.26. In the figure below we could see the graph of y = 1 - x 2 , as well as some transformations of it. More precisely: the graph of y = 1 - (2 x ) 2 is the graph of y = 1 - x 2 shrinked horizontally by a factor of 2; the graph of f ( x ) = 1 - ( x/ 2) 2 is the graph of y = 1 - x 2 stretched horizontally by a factor of 2.
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36 J. S´ anchez-Ortega Example 2.27. Given the graph of y = x , use transformations to graph y = x - 2, y = x - 2, y = - x , y = 2 x , and y = - x . Solution: The graph of the square root function y = x is shown in Figure (a). In the other parts of the figure we sketch y = x - 2 by shifting 2 units downward, y = x - 2 by shifting 2 units to the right, y = - x by reflecting about the x -axis, y = 2 x by stretching vertically by a factor of 2, and y = - x by reflecting about the y -axis.
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2. Functions 37 2.10 More examples of piecewise-defined and absolute value functions Example 2.28. Consider the function f ( x ) = | 1 + x | - 1 x Determine the domain of f , express it as a piecewise-defined func- tion and sketch its graph.
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