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Scaling & reflecting absolute value functions: graph

The graph of y=k|x| is the graph of y=|x| scaled by a factor of |k|. If k<0, it's also reflected (or "flipped") across the x-axis. In this worked example, we find the equation of an absolute value function from its graph.

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  • leaf green style avatar for user alexander
    How would you stretch it?
    (5 votes)
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    • mr pink green style avatar for user David Severin
      You always stretch any function by adding an a in front of highest power of the function, so with the absolute value parent function, f(x) = | x |, adding a number greater than 1 causes a vertical stretch such as f(x) = 2 | x |, and between 0 and 1 is a vertical compression such as f(x) = 1/2 | x |
      (5 votes)
  • leafers ultimate style avatar for user jacksonlc
    Would you recommend stretching the function or flipping the function first?
    (5 votes)
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  • blobby green style avatar for user lopezpablo044
    So, basically, y= -4 lxl is the equation. Would you say, in general of course, that -4, when outside the abs. value symbol, is kind of like the slope (-4/1)?
    (2 votes)
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  • blobby green style avatar for user Hunter Davis
    I am confused about scaling using the factor. I understand about shifting to the left or to the right, and going up or down on the y axis, but I don't understand how you determine to stretch or compress the graph with the factor. For example, f(x) = 2|x+3| +2, please explain how to scale it by 2. The |x+3| means move the whole graph to the left by 3; the +2 means move the graph up by 2 on the y axis. But how do I scale it to a factor of 2? Thank you
    (2 votes)
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    • leaf green style avatar for user Nikolay
      Hunter,

      Have you graphed linear equations before? If so, you should be familiar with having a number in front of x.

      E.g. y = 2x. For every x, y is twice as much. The line becomes quite steep when you have a whole number; a fraction causes the line to be less steep.

      Same thing here. y = 2|x| has a compressed V-shape; y = 1/2|x| is stretched out.

      In your equation, start off with y = |x|. Before doing the shift left and shift up stuff, draw y = 2|x|. You will see a compressed V-shape compared to y = |x|. Then apply the other transformations.
      (1 vote)
  • leaf green style avatar for user Ms.Tajbibi Shamim
    Why not -5 times as it touch both lines?
    (2 votes)
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  • blobby green style avatar for user Angelina Malkasian
    If it is across the x axis, how could one change the y- intercept if the slope is not determined?
    (2 votes)
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    • mr pink green style avatar for user David Severin
      While I am not sure exactly what you are asking, the problem shows the vertex of the absolute value function at the origin. If the vertex is anywhere else but the vertex, then the y-intercept would have to be the additive inverse of the original y-intercept. On a graph, it should not be too difficult to determine the slope. Is this sort of what you are asking?
      (0 votes)
  • piceratops ultimate style avatar for user Iaroslava Mykhalchuk
    whats would you do if you had a fraction?
    (2 votes)
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    • leaf green style avatar for user Nikolay
      A fraction causes the V-shape of the graph to stretch out (whole number, as in this video, causes the graph to squeeze).

      Let's compare y = |x| and 1/2|x|

      |x|: When x=1, y=1; when x=-1, y=1

      1/2|x|: When x=1, y=1/2; when x=-1, y=1/2

      If you have graph paper, try drawing these graphs and see what the fraction does.
      (0 votes)
  • duskpin ultimate style avatar for user Haha5X
    What is the difference between a horizontal stretch and a vertical stretch? Don't they still look the same??
    (1 vote)
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  • duskpin ultimate style avatar for user Reda Haji
    so basically the 4 is considered as a slope ?
    (1 vote)
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  • blobby green style avatar for user keckerson.student
    why didn't you just say reflect the shape a crossed the x access?
    (1 vote)
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Video transcript

- [Instructor] Function G can be thought of as a stretched or compressed version of F of X is equal to the absolute value of X. What is the equation for G of X? So you can see F of X is equal to the absolute value of X here in blue, and then G of X, not only does it look stressed or compressed, but it also is flipped over the X axis. So like always, pause this video and see if you can up yourself with the equation for G of X. Alright, now let's work through this together. So there's a couple of ways we could do it. We could first try to flip F of X, and then try to stretch or compress it, or we could stretch or compress it first, and then try to flip it. Let's actually, let's flip it first, so let's say that we have a function that looks like this. It's just exactly what F of X is, but flipped over the X axis. So it's just flipped over the X axis, so all the values for any given X, whatever Y you used to get, you're not getting the negative of that. So this graph right over here, this would be the graph. I'll call this, Y is equal to the negative absolute value of X. Whatever the absolute value of X would have gotten you before, you're now going to get the negative of the opposite of it. So this is getting us closer to our definition of G of X. The key here is how do we appropriately stretch or squeeze this green function? So let's think about what's happening. On this green function, when X is equal to one, the function itself is equal to negative one, but we want it, if we want it to be the same as G, we want it to be equal to negative four. So it's actually four times the value. For a given X, at least for X equals one, G is giving me something four times the value that my green function is giving. That's not just true for positive Xs. It's also true for negative Xs. You can see it right over here. When X is equal to negative one, my green function gives me negative one, but G gives me negative four. So it's giving me four times the value. It's giving me four times the negative value, so it's going even more negative, so what you can see, to go from the green to G, you have to multiply this thing right over here by four. So that is what essentially stretches it down, stretches it down in the vertical direction. So we could say that G of X is equal to, it's not negative absolute value of X, negative four times the absolute value of X. And you could have done it the other way. You could have said, "Hey, let's first stretch "or compress F." And say, alright, before we even flip it over, if we were to unflip G, it would look like this. If we were to unflip G, it would look like this. If were to unflip G, so this thing right over here, this thing looks like four times F of X. We could write this as Y is equal to four times F of X, or you could say Y is equal to four times the absolute value of X, and then we have a negative sign. Whatever positive value you were getting before, you now get the opposite value, and that would flip it over and get you to G, which is exactly what we already got.