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## College Algebra

### Course: College Algebra > Unit 11

Lesson 1: Graphs of absolute value functions# Shifting absolute value graphs

CCSS.Math:

The graph of y=|x-h|+k is the graph of y=|x| shifted h units to the right and k units up. See worked examples practicing this relationship.

## Want to join the conversation?

- I broke my replay button and I still don't understand this! Why move 3 to the right and write x-3 when x+3 seems to make sense?(29 votes)
- If you are trying to find where the new "0" would be, when x = 3, then x-3 =0, so this would move it to the right. for x + 3, you would have to add -3 (to the left) to get 0. The point slope form shows this best when you have y - y1 = m(x-x1). In this case, a positive y1 actually would move the graph down and a negative y1 would move it up, but when we see it in point slope form y = mx + b, the b stays what it is (positive up and negative down).(38 votes)

- How the equation would be if we shift it only up 4?(12 votes)
- The answer would've been y=|x|+4(3 votes)

- Why would you subtract 3 if your are going to the right of the graph on the X-Axis instead of adding? So would that mean you would add if you were going the left of the X-Axis?5:35(18 votes)
- I think you lack lifting y=x graph and need doing some practice or review.

When you subtract 3 you're shifting down 3 points on the Y-Axis

and

When you add 4 you're shifting up 4 points on the Y-Axis

OR

When you subtract 3 you're shifting right 3 points on the X-Axis

and

When you add 4 you're shifting left 4 points on the X-Axis

---

combining the both of point of views is NOT correct like you did:

When you subtract 3 you're shifting down 3 points on the Y-Axis

and

When you add 4 you're shifting left 4 points on the X-Axis(3 votes)

- Ummm... I think we use absolute value to find the distance from 0 to the number itself. Am I correct?(3 votes)
- Yes, you are
**absolutely**correct. Absolute value finds the distance of a number from 0. For example, the absolute value of -3 is 3, but the absolute value of 99 is 99. Hope this helps!(4 votes)

- wait.. so uhh can we like simplify

y = |x - 3| + 4

to

y = |x - 3 + 4|

and then simplify that to

y= |x + 1|(3 votes)- Those are not the same equation.

y = |x-3|+4 has its vertex point at (3,4)

y = |x+1| has its vertex point at (-1,0)

Hope this helps.(6 votes)

- Hi! I am currently taking Algebra 1 on Khan Academy, but I feel like I missed something. Could someone please point me to where Sal first talked about absolute value?(3 votes)
- You should first do the Get ready for Algebra 1 course first :)(3 votes)

- What happens when you shift to the left?(2 votes)
- Shifting three to the left would be the equation y = | x + 3 |(5 votes)

- I am a bit confused (1:40). What is Khan talking about when you switch signs? Why do you -3 to move it to the right? Is there is a way to explain this without using an example of plotting points?(1 vote)
- When he mentions switching signs he means what is inside of the abslute value signs. Let's first lok at just |x|.

|-1| = 1 and all other negative numbers are turned to positive. When you graph the absolute value function it makes a sudden sharp turn when you get to 0, which in other words is saying when you SWITCH SIGNS fro the negative numbers to non negative, the graph turns.

Why it moves 3 to the right is because you can move graphs around. You can move it up, down, left, and right. You just have to change the equation.

Specifically to move a graph to the right you need to determine the inside of the function. absolute value is pretty easy. inside the function is inside the absolute value bars. Once you find the inside of the function you just need to subtract a number from the variable to move right. so |x-1| goes to the right one. |x-2| goes tot he right two, and so on. if you add you go left, so |x+3| goes to the left 3.

If you are asking why it moves like that when you add or subtract then that is a little more tricky to answer. My suggestion is to think backwards with an answer and what youw ould need to change.

|x| has the angle at x=0. let's say we wanted it at 5 instead. We know we have to add or subtract something inside to make it happen. sothat means we will have x=5 and y = 0

|5+a|=0 So what does a have to be here? obviously -5 so that tells us |x-5| has the 0 point now at x=5, or in other words the graph was moved 5 places to the right.

Does that make sense?(6 votes)

- I kind of find it confusing.(3 votes)
- So is the white line always a V like that?(1 vote)
- In absolute value functions, the line it forms is always a V. It is not necessarily the same angle, and the vertex may be in a different spot, but it is a V nonetheless.(5 votes)

## Video transcript

- [Instructor] This right over here is the graph of y is equal
to absolute value of x which you might be familiar with. If you take x is equal to negative two, the absolute value of
that is going to be two. Negative one, absolute value is one. Zero, absolute value is zero. One, absolute value is one. So on and so forth. What I wanna do in this video is think about how the
equation will change if we were to shift this graph. So in particular, we're
gonna first think about what would be the equation
of this graph if we shift, if we shift three to the right and then think about how that will change if not only do we shift three to the right but we also shift four up, shift four up, and so once again pause this
video like we always say and figure out what would the equation be if you shift three to
the right and four up? Alright, now let's do this together. So let's just first
shift three to the right and think about how that
might change the equation. So let's just visualize what
we're even talking about. So if we're gonna shift
three to the right, it would look like, it would look like this. So that's what we first wanna
figure out the equation for and so how would we think about it? Well, one way to think about it is, well, something interesting is happening right over here at x equals three. Before, that interesting
thing was happening at x equals zero. Now, it's happening at x equals three. And the interesting
thing that happens here is that you switch signs
inside the absolute value. Instead of taking an
absolute value of a negative, you're now taking the absolute value as you cross this point of a positive and that's why we see a
switch in direction here of this line and so you see the same thing
happening right over here. So at this point right over here, we know that our function, we know that our equation
needs to evaluate out to zero and this is where it's
going to switch signs and so we say, okay, well, this
looks like an absolute value so it's going to have the form, y is equal to the absolute
value of something and so you say, okay, if x is three, how do I make that equal to zero? Well, I can subtract three from it. If I say y is equal to the
absolute value of x minus three, well, let's try it out. Let's see if it makes sense. So when x is equal to three, three minus three is zero, absolute value of that is zero. That works out. When x is equal to four, four minus three is one,
absolute value of one is indeed, is indeed one. And if x is equal to two, well, two minus three is negative one but the absolute value of that is one. So once again, I'm showing you this by really trying out numbers, trying to give you a
little bit of an intuition because that wasn't obvious to
me when I first learned this that if I'm shifting to the right which it looks like I'm
increasing an x value but what I would really do is replace my x with an x minus the amount
that I'm shifting to the right but I encourage you to try numbers and think about what's happening here. At this vertex right over here, whatever was in the absolute
value sign was equaling zero. It's when whatever was in
the absolute value sign is switching from negative
signs to positive signs. So once again, if you
shift three to the right, that has to happen at x equals three. So whatever is inside
the absolute value sign has to be equal to zero at
x equals three and this, pause this video and
really think about this if it isn't making sense and even as you get more
and more familiar with this, I encourage you to try out the numbers. That will give you more,
instead of just memorizing, hey, if I shift to the right, I replace x with x minus
the amount that I shift. Always try out the numbers
and try to get an intuition for that why that works. But now, let's start from
here and shift four up and shifting four up is in
some ways a lot more intuitive. So let me do that. Let me shift four up. Alright, so let me move. I'm gonna go up one, two, three and four. I think I got that right. So now I've shifted four up. And just as a reminder
of what we've even done, the first part we shifted
three to the right and now we are shifting four up. So now, now we are shifting four up. So before, y equaled zero here but now, y needs to be equal to four. So whatever this was evaluating, do we now have to add four to it? So when we just shifted
three to the right, our equation was y is
equal to the absolute value of x minus three but now whatever we were getting before, we now have to add four to it. We're going up in the vertical direction. So we just have to add four. Now, this makes a little bit more sense. If you're shifting in
the vertical direction, if you shift up in the vertical direction, well, you just add a constant
by the amount you're shifting. If you shift down in
the vertical direction, well, you would subtract. If we said shift down four, you would subtract four right over here. The less intuitive thing
is what we did with x 'cause when you shift to the right, you actually replace your x with x minus the amount that you shifted but once again, try out numbers until it really makes some
intuitive sense for you but this is what we would finally get. The equation of this thing right over here is y is equal to the absolute value of x minus three plus four.