ABS(f(x)) and f(ABS(x)) GraphsDate: 08/27/2013 at 14:44:55 From: Pavan Patel Subject: Determining the graphs of functions A graph of f(x) has a hump up to y = 1 for x between 0 and 2. To the left of the origin, it slowly tails off to negative infinity. I have to sketch the graphs of the functions (a) f(x + 1) (b) f(-x) (c) abs(f(x)) (d) f(abs(x)) I am confused as to how I am supposed to plot f(abs(x)): even after I looked up its solution, I am unsure as to how they got that answer. Date: 08/27/2013 at 15:31:24 From: Doctor Peterson Subject: Re: Determining the graphs of functions Hi, Pavan. Let's draw your graph like this: +1 * | * * |* * +-----+-----*-----+-----* -2 -1 *| 1 2 * | * +-1 (a) f(x + 1) will be a horizontal shift, 1 unit to the left: *1 * | * * | * +-----+-----*-----+-----*-----+ -3 -2 *-1 | 1 2 * | * +-1 To check this, what will f(x + 1) be for x = 0? f(0 + 1) = f(1), which from the given graph is 1. And that's what the shifted graph shows. For more on this, see the prior Dr. Math conversation Graph with f(x) http://mathforum.org/library/drmath/view/54509.html (b) f(-x) will be a reflection in the y axis, because x is replaced by -x for any point on the graph: * +1 * * | * *| *-----+-----*-----+-----+ -2 -1 |* 1 2 | * +-1 * To check this, what will f(-x) be for x = -1? f(-(-1)) = f(1), which from the given graph is 1. And that's what the reflected graph shows. For more on this, see Order of Transformations of a Function http://mathforum.org/library/drmath/view/68503.html (c) |f(x)| will reflect any negative parts of the graph up over the x-axis, making them positive: * +1 * * | * * *|* * +-----+-----*-----+-----* -2 -1 | 1 2 | +-1 To check this, what will |f(x)| be for x = -2? |f(-2)|, which from the given graph is |-1| = 1. And that's what the new graph shows. For more on this, see Graphing Absolute Values http://mathforum.org/library/drmath/view/60968.html Graphing the Absolute Value/Square Root of a Function http://mathforum.org/library/drmath/view/61112.html (d) f(|x|) will ignore values of f for negative x (since the argument of f will never be negative, and replace it with a reflection of the right side, as explained in the fifth page above: * +1 * * * | * * * *|* * *-----+-----*-----+-----* -2 -1 | 1 2 | +-1 To check this, what will f(|x|) be for x = -1? f(|-1|) = f(1), which from the given graph is 1. And that's what the new graph shows. For more on this, see Graphing f(2x) and f(|x|) http://mathforum.org/library/drmath/view/64038.html I hope you notice how I checked my graphs; this is very good thing to do both to make sure you're right and to get a better understanding of what these transformations do, and why. Ideally, you should check more than one point, especially in the absolute value examples. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 08/27/2013 at 15:47:14 From: Pavan Patel Subject: Thank you (Determining the graphs of functions) Thank you! Your explanation was really thorough and helpful! |
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