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### Course: Algebra 1 > Unit 10

Lesson 1: Graphs of absolute value functions- Shifting absolute value graphs
- Shift absolute value graphs
- Scaling & reflecting absolute value functions: equation
- Scaling & reflecting absolute value functions: graph
- Scale & reflect absolute value graphs
- Graphing absolute value functions
- Graph absolute value functions
- Absolute value graphs review

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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.

## Want to join the conversation?

- How would you stretch it?(5 votes)
- You could also think of it this way. When we have the function f(x) = |x|, it's also the same as f(x) = 1|x|, where the 1 there is the gradient of the function on a graph.

Remember, gradient is the change in y over the change in x, in this case, the gradient is 1/1 (which is 1) in other words, every change of 1 y (or every time a point on the graph moves 1 in the y direction), there will be a change of 1 x (it will also have to move 1 in the x direction).

Imagine we wanted to stretch it along the vertical direction (y direction). That means the change in y would have to be greater than the change in x. For example, if the function was y = 2|x|, the gradient was 2, or 2/1, which means if the point move 2 in the y direction, it would have to move 1 in the x direction. If you graph the function, it will look stretched. All you need to do is changing the**gradient**of the function.

Am I making myself clear?(7 votes)

- Would you recommend stretching the function or flipping the function first?(5 votes)
- It's irrelevant and completely up to your preference.(5 votes)

- is -|4x| also a correct solution?(5 votes)
- Yes, g(x) = -|4x| would work as well.(3 votes)

- i don't get it 🔥(3 votes)
- What do you need help with?(3 votes)

- why are we using absolute values?(3 votes)
- Would you recommend stretching the function or flipping the function first?(2 votes)
- it doesn't matter you can do either(1 vote)

- So if you have f(x)=4|x|, does that just mean that instead of the coordinate at (1,1), it's (1,4) instead?(2 votes)
- What is the difference between a horizontal stretch and a vertical stretch? Don't they still look the same??(1 vote)
- No, stretching is like pulling either up (vertical) or out (horizontal). A vertical compression pushes things toward the x axis, so a vertical compression will look the same as a horizontal stretch, and a vertical stretch will look like a horizontal compression.(2 votes)

- Why not -5 times as it touch both lines?(1 vote)
- 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)?(1 vote)
- Kind of...

A simple absolute value function like you have will create a V-shaped graph. The -4 does 2 things to the V.

1) It makes the V narrower (like having a steeper slope

2) The negative sign flips the V upside down.

Hope this helps.(1 vote)

## 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.