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## 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

CCSS.Math:

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 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)

- Would you recommend stretching the function or flipping the function first?(5 votes)
- It's irrelevant and completely up to your preference.(4 votes)

- 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)
- 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.(2 votes)

- 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)
- 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)

- Why not -5 times as it touch both lines?(2 votes)
- If it is across the x axis, how could one change the y- intercept if the slope is not determined?(2 votes)
- 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)

- whats would you do if you had a fraction?(2 votes)
- 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)

- 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)

- so basically the 4 is considered as a slope ?(1 vote)
- Sort of yes. To the right of the vertex, the slope is 4 and to the left of the vertex the slope is -4.(1 vote)

- why didn't you just say reflect the shape a crossed the x access?(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.