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Shifting parabolas

The graph of y=(x-k)²+h is the resulting of shifting (or translating) the graph of y=x², k units to the right and h units up. For example, y=(x-3)²-4 is the result of shifting y=x² 3 units to the right and -4 units up, which is the same as 4 units down.

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  • duskpin sapling style avatar for user Tianze
    For instance y=(x±9)^-1
    Why does the sign before 9 have a counter effect on the parabola?
    (6 votes)
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    • blobby green style avatar for user turtlefan69xo
      wait, do you mean y=(x±9)^2 - 1? There is no squared value in the original question, just ^-1. Anyways, assuming that you mean y=(x±9)^2 - 1, then I would be happy to tell you how I think of the counter effect, as you put it. I pretend that I am trying to balance something on a seesaw, where the 0 is the pivot point. Imagine that you had a friend who weighed 9 kilos more than you. Your friend is x+9, and you are x. If you and your friend want to balance, you must shift the seesaw in your direction, or the heavier friend will tip it over. Basically, +9 means that it is 9 points too heavy on the positive side, so if the positive side is too heavy, what do you have to do? You have to shift the whole system to the left, so it can still balance. I hope this helps! I also hope that people still know what a seesaw is, even though people don't seem to play outside anymore.
      (19 votes)
  • duskpin ultimate style avatar for user grigor21
    y=(x-h)^2+k How do negative values of h represent leftward shifts? I cannot get this one, Sal in the video explained that when we shift h units to the right we substract h units from the function.
    (2 votes)
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    • mr pink green style avatar for user David Severin
      This is going to be true for all functions, so lets start with a linear equation y = x + 3. the y intercept is 3 (set x=0) and the x intercept is -3 (set y = 0). If we keep it as a change in y, we have y = x + 3, so it is easy to see the y intercept. By "making it a change in x" instead, we show it as y = (x + 3) + 0. So if we put in a negative 3 for x, we get y = 0 which gives us the correct x intercept. So we had to have the opposite sign for a change in x.
      Similarly for the quadratic function such as y = (x + 3)^2 + 5, we would have to set x = -3 in order to make what is inside the parentheses to be 0, we have to change the sign.
      SO a change in y follows the sign, a change in x has to be the opposite sign.
      (5 votes)
  • blobby green style avatar for user Tofunmi Adewumi
    How would you do this? Shifting f(x) 1 unit right then 2 units down. The equation is f(x)=x^2-2x-1.

    My book says this

    to shift 1 unit right its

    g(x)=(x-1)^2-2(x-1)-1
    =x^2-4x+2

    for shift 2 units down

    h(x)=x^2-2x-1-2
    =x^2-2x-3

    I need some help understanding this please
    (2 votes)
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    • mr pants purple style avatar for user The Purple Bear
      Translations are often confusing at first glance. You just kind of have to memorize it.
      Here are the basics.
      Horizontal Translations (moving along x axis)
      If we have f(x),
      f(x-1) is the function moving to the RIGHT by 1.
      f(x+1) is the function moving to the LEFT by 1.
      confusing, I know
      Vertical Translation (moving along y axis)
      f(x)
      f(x)+1 is the function moving UP by 1.
      f(x)-1 is the function moving DOWN by 1.
      Now how do we use these?
      Let's say we have
      f(x)=3x+5
      and we want to move it to the right by 4 units.
      that would make it f(x-4).
      Replace all the x's inside the function with x-4.
      So you get 3(x-4)+5
      This equals to
      3x-12+5=3x-7.
      So 3x-7 is what we get when we move the function to the right by 4 units. *graph it and check!*
      Let's say we have the same equation, f(x)=3x+5
      but we want to make it go down by 3!
      Remember,
      f(x)-3 means it goes down by 3
      f(x)-3=(3x+5)-3
      f(x)-3=3x+2
      The new function is 3x+2, which is the function where we moved it down by 3!
      (5 votes)
  • blobby green style avatar for user ariel.nawy
    would it be right to write it down like this?: g(x)+4=(x-3)squared
    (3 votes)
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  • leaf green style avatar for user Sally
    So just to be clear:

    Shifting to the right by 3 is (x-3)
    Shifting to the left by 3 is (x+3)

    Shifting down by 3 is (x)-3
    Shifting up by 3 is (x)+3
    (2 votes)
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  • blobby green style avatar for user White, Kennedy
    Does anyone know the mentioned videos that explain shifting more in depth?
    (2 votes)
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  • cacteye yellow style avatar for user Br Paul
    If moving the vertex to the right makes it (x-3), why, when I move the vertex down four, doesn't the equation then equal (x-3)+4?
    (1 vote)
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    • mr pink green style avatar for user David Severin
      If you have y = 2(x-5)^2 + 2, the 5 is with the x, so if you want to do the same with the ys, you have to subtract 2 on both sides to get y - 2 = 2(x-5)^2, in this case the y would also have to change signs (similar to the point slope form of a linear equation y-y1=m(x-x1). We could do the same thing with this, y = m(x-x1)+y1 where x1 changes sign and y1 would stay the same, So when the 2 is on the same side as the x (right side of equation), you do not change the sign. Hope this makes sense.
      (4 votes)
  • duskpin sapling style avatar for user Anna
    if you minus by a number when you shift to the right, do you add by a number when you shift to the left?
    (1 vote)
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    • mr pink green style avatar for user David Severin
      Yes that is correct. If you have something like (x-5)^2 + 3, that negative shifts to the right because you need to have x=5 for the inside of parentheses to be 0 (5-5)^2 and if you have (x + 4)^2 - 3, you need to have x=-4 to had to have it be 0 because (-4+4)^2=0.
      (4 votes)
  • blobby green style avatar for user Arbaaz Ibrahim
    How is y=f(x-3) and y=(x-3)^2 the same?
    At about minutes into the video.
    (2 votes)
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  • duskpin seed style avatar for user mareli vaneti
    Does it matter if we write y + 4 = (x-3)^2 instead of y = (x-3)^2 - 4 ? Should be the same but is one preferred over the other? Last video he used the former format, where the 4 was being added to the y.
    (1 vote)
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Video transcript

- [Instructor] Function g can be thought of as a translated or shifted version of f of x is equal to x squared. Write the equation for g of x. Now, pause this video, and see if you can work this out on your own. All right, so whenever I think about shifting a function, and in this case, we're shifting a parabola, I like to look for a distinctive point. And on a parabola, the vertex is going to be our most distinctive point. And if I focus on the vertex of f, it looks like if I shift that to the right by three, and then if I were to shift that down by four, at least our vertices would overlap. I would be able to shift the vertex to where the vertex of g is. And it does look, and we'll validate this, at least visually, in a little bit, so I'm gonna go minus four in the vertical direction, that not only would it make the vertices overlap, but it would make the entire curve overlap. So we're going to make, we're gonna first shift to the right by three. And we're gonna think about how would we change our equation so it shifts f to the right by three, and then we're gonna shift down by four. Shift down by four. Now, some of you might already be familiar with this, and I go into the intuition in a lot more depth in other videos. But in general, when you shift to the right by some value, in this case, we're shifting to the right by three, you would replace x with x minus three. So one way to think about this would be y is equal to f of x minus three, or y is equal to, instead of it being x squared, you would replace x with x minus three. So it'd be x minus three squared. Now, when I first learned this, this was counterintuitive. I'm shifting to the right by three. The x-coordinate of my vertex is increasing by three, but I'm replacing x with x minus three. Why does this make sense? Well, let's graph the shifted version, just to get a little bit more intuition here. Once again, I go into much more depth in other videos here. This is more of a worked example. So this is what the shifted curve is gonna look like. Think about the behavior that we want, right over here, at x equals three. We want the same value that we used to have when x equals zero. When x equals zero for the original f, zero squared was zero. Y equals zero. We still want y equals zero. Well, the way that we can do that is if we are squaring zero, and the way that we're gonna square zero is if we subtract three from x. And you can validate that at other points. Think about what happens now, when x equals four. Four minus three is one squared. It does indeed equal one. The same behavior that you used to get at x is equal to one. So it does look like we have indeed shifted to the right by three when we replace x with x minus three. If you replaced x with x plus three, it would have had the opposite effect. You would have shifted to the left by three, and I encourage to think about why that actually makes sense. So now that we've shifted to the right by three, the next step is to shift down by four, and this one is little bit more intuitive. So let's start with our shifted to the right. So that's y is equal to x minus three squared. But now, whatever y value we were getting, we want to get four less than that. So when x equals three, instead of getting y equals zero, we want to get y equals four less, or negative four. When x equals four, instead of getting one, we want to get y is equal to negative three. So whatever y value we were getting, we want to now get four less than that. So the shifting in the vertical direction is a little bit more intuitive. If we shift down, we subtract that amount. If we shift up, we add that amount. So this, right over here, is the equation for g of x. G of x is going to be equal to x minus three squared minus four. And once again, just to review, replacing the x with x minus three, on f of x, that's what shifted, shifted right by three, by three. And then, subtracting the four, that shifted us down by four, shifted down by four, to give us this next graph. And you can visualize, or you can verify visually, that if you shift each of these points exactly down by four, we are, we are indeed going to overlap on top of g of x.