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Transforming the square-root function

Sal shows various examples of functions and their graphs that are a result of shifting and/or flipping y=√x. Created by Sal Khan.

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  • leaf red style avatar for user Shyowls
    What if it 's like 2x+3? with that radical sign over it ? How would you graph that?
    (4 votes)
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    • purple pi pink style avatar for user Sachin
      If you have the function y = sqrt(2x+3), you can rewrite the right hand side of this as:
      y = sqrt(2(x+3/2)).

      Then, using the property that sqrt(ab) = sqrt(a)*sqrt(b), you can rewrite this again as:
      y = sqrt(2) * sqrt(x+3/2).

      Now, notice that sqrt(2) is no more than a constant, you all you've done is stretched the graph vertically byu a factor of sqrt(2). Then, notice that under the second radical sign, you've got a shift to the left by 3/2. To show how this process makes sense, try graphing both y = sqrt(2x+3) and y = sqrt(2) * sqrt(x+3/2). You should get the same thing.

      To graph it, know what the graph of y = sqrt(x) looks like first (its a parabola on its side with only the top half). Then, notice that you've shifted the graph to the left by 3/2 and stretched the entire graph by sqrt(2). All done!
      (13 votes)
  • leaf blue style avatar for user Zaid Dinally
    Sal writes sqrt - (x + 3)

    I can see the principle of reflection this radical function along line of x = -3. However will this not yield complex numbers? Thank you
    (4 votes)
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    • mr pink red style avatar for user andrewp18
      It will not yield imaginary numbers as long as "x" is chosen carefully. We can find exactly for which values of x no complex numbers result. We do this by finding the domain of the function:
      ƒ(x) = √[-(x + 3)]
      The radicand must be greater than or equal to 0 in order for the function to yield only real numbers:
      -(x + 3) ≥ 0
      -x - 3 ≥ 0
      -x ≥ 3
      x ≤ -3
      Therefore, all values of "x" that produce real numbers in the aforementioned equation are all numbers lesser than or equal to -3. This is also why the graph of the function starts at -3 on the x-axis and keeps going in the negative direction, but does not go in the positive direction at all. Comment if you have questions.
      (5 votes)
  • leafers seed style avatar for user rylee foster
    Why am I not getting any of this.
    (4 votes)
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  • female robot grace style avatar for user Jazmyn
    Another way to say why x is positive instead of negative, you could say that y=√x -> or x=y^2. Because you want to move x three spaces to the left, you would do x=y^2 -3. Then, to convert it back, it would become y^2=x+3, or y=√x+3. idk just another way to understand it I guess.
    (2 votes)
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  • blobby green style avatar for user Kashif Ahmad
    You said you have to make sure that whatever is under the radical will equal 0 at the origin like if we want to shift 2 to left we +2 to make sure that the number under radical is zero,but i want to ask why is that necessary to make it 0.?
    (2 votes)
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  • leafers seedling style avatar for user Joseph
    how do you compute √-(x+3) ? I thought you couldn't have negatives under the radical
    (2 votes)
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  • duskpin tree style avatar for user Jill K
    does anyone know a video or link that describes how to find the x intercept of a transformation of the function y = cube root of x^2?
    (1 vote)
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  • male robot hal style avatar for user King Henclucky
    What would the graph of y = 3* sqrt(x) or y = sqrt(3x) or y = -3*sqrt(x), etc.?
    (1 vote)
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    • orange juice squid orange style avatar for user camcam620
      y=3*sqrt(x) would be the a slightly steeper version of y=sqrt(x). Think of it this way: each value for sqrt(x) is multiplied by three, making it it much steeper.

      y=-3*sqrt(x) would be the exact mirror along the y axis of y=3*sqrt(x).

      Hope this helped.
      (2 votes)
  • hopper cool style avatar for user Tran Ngoc Nam Vu
    At , what is y when sqrt(x+3) is negative? I don't understand!
    (1 vote)
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    • blobby green style avatar for user InnocentRealist
      sqrt(x) means positive (principal) sqrt always. If someone wants the negative sqrt they must write -sqrt(x). This is a rule.

      When -(x+3) is negative (that is, to the right of -3), y = sqrt(-(x+3)) is undefined. At -3, -(x+3) is 0. As x moves to the left of -3, -(x+3) increases from 0, and so its sqrt, y, increases.
      (1 vote)
  • blobby green style avatar for user Brooke Wilson
    When you graph a radical function how do you tell whether the x-value is negative or positive? I get that the y-value maintains it's negativity/positivity from the equation but the x seems to switch at random.
    (1 vote)
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Video transcript

So let's think about the graph of y is equal to the principal root of x. And then we'll start playing around with this and see what happens to the graph. So y is equal to the principal root of x. Well, this is going to be undefined if we want to deal with real numbers. For x being any negative value. So the domain here is really x is greater than or equal to 0. When x is 0, y is going to be equal to 0. When x is 1, the principal root of 1 is positive 1. So it's going to be like that. When x is 4, the principal root of 4 is 2. When x is 9, the principal root of 9 is 3. So this is what it is going to look like. Which is going to look more like this. So it's going to look something like that. That's my best attempt at actually graphing it. Now let's think about what happens if we wanted to shift it in some way. So let's say we wanted to shift it up. Let's say we wanted to shift it up by 4. So how would we do that? Well, whatever value we're getting here, we want y to be 4 higher. So we could just add 4 to it. So we could just use y is equal to the square root of x plus 4. So that would be like taking this graph right over here. So let me copy and then let me paste it. So it's like taking this graph and we're shifting it up 1, 2, 3, 4. It would look that. Well, that was pretty straightforward. Let me do it in that same blue color so that you recognize that that's that one right over there. But what if we wanted to shift it, let's say, to the left. Let's say we wanted to do something like this. Let's say we wanted to shift it to the left by 3. Like that. So how would we define the function then? And I'll do this in this orange color. We want to shift by 3. So you're shifting to the left by 3. So think about it. This point, y equaled 0, right over here, where whatever you put under the radical was equal to 0. So what do you have to put under the radical here to get 0? Well, here x is negative 3. So if you put x plus 3 under the radical then you are going to get-- and you take the square root of that-- you're going to get 0. So this right over here, this orange function, that is y. Let me do it over here. y is equal to the square root of x plus 3. And once again it might be counter-intuitive. We went from square root of x to square root of x plus 3. When we added 4 outside of the radical that shifted it up. But when we add 3 inside of the radical instead of it shifting it to the right, instead of shifting it that way, it shifted it to the left. It made this point go from 0 to negative 3. And the important thing to realize is what makes y equal to 0. Over here y equals 0 when x is 0. Over here and over here, y is equal to 0 when x plus 3 is equal to 0, or x is equal to negative 3. And you could do that for other points to see that it does definitely shift to the left. And that's an important thing to realize. This isn't actually just for radical functions. This is actually for functions in general. If you're just going to add a 4-- add a number out here-- whatever you add is going to shift it up or down. If this was a minus 4 it would have shifted it down. But within, when you replace the x with an x plus 3, when you replace it with an x plus 3, this actually shifted it to the left by 3. If you wanted to shift it to the right by 3 you would put an x minus 3 over there. Well, that's all interesting. But let's say that I wanted to flip this thing over. So I wanted it to look like this. Let me see if I can draw it. I want this graph to look something like that. I'll try my best to-- to look something like that. So it's flipped around. I could do a better job than that. So we have that point on it. And then we're going to go 3 and then we're going to have that point on it. So I want it to look-- actually I did a decent job the first time I drew it-- I want to look something like that. So essentially I want to take its mirror image around the line x is equal to 3. How could I do that? Well, now my domain is different. Now my domain-- it should be undefined for anything where x is greater than negative 3. And it should be defined for any x that is less than or equal to negative 3. Or another way to think about it is we need to flip the sign of whatever we have under the radical. So this thing over here-- let me actually scroll over a little bit-- this thing over here in green could be y is equal to the square root of the negative of x plus 3. And I encourage you to try some x values here to try it out. What we've done is we've essentially flipped what happens under the radical. Now in order to get a positive value under the radical, now x plus 3 has to be negative. And the only way that x plus 3 is negative, or the only way that x plus 3 is, I guess you can say nonpositive, is if x is less than negative 3. Now what if we wanted to do something even more interesting? What if we wanted to flip this one right over here over the horizontal axis, over y equals 0? Well, then we're just flipping what root we take. So that would be y is equal to the negative square root of negative x plus 3. So it would look like this. It would look like that. And if we wanted to shift that thing, we could just add or subtract something outside of the radical. So let's say we wanted to do-- let me copy and paste this. So let's copy and then let's see. Let's say that we wanted to shift it down here. So instead of beginning at y equals 0 right over here, where y equals negative 4, then we would just subtract 4 outside of the radical. So this thing and this thing-- I'm running out of colors here-- this thing right over here would be y is equal to negative square root of negative x plus 3 minus 4. And so we could keep going on and on and on. But hopefully this gives you a sense of the different ways you could manipulate this thing. And we'll do some more examples to get a better understanding of it.