If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Algebra 1 (FL B.E.S.T.)>Unit 6

Lesson 5: Graphs of absolute value functions

# Scaling & reflecting absolute value functions: equation

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 a description of the transformation performed on y=|x|.

## Want to join the conversation?

• Is there any way to tilt or rotate the function? I.e rotate the function by say 45 degrees.
• yes, there is a way, but it requires knowledge of things called parametric equations, trigonometry, and also linear algebra. you'll encounter these later on in your math classes (assuming you take them past grade 10)
• The 7 is the slope, right?
• Sort of, but not quite. The function would be f(x) = - 7 | x|. For x ≤ 0, the slope would be 7 with a y intercept of (0,0). For x ≥ 0, the slope would be -7 with a y intercept of (0,0). So you have to consider which part of the graph you are talking about.
• what scaled even mean?
• "Scaled" means the graph rises faster or slower than the standard function of y=|x|. For example, y=2|x| rises twice as fast so the V-shape will be narrower.
Hope this helps.
• can functions have graphs in a v shape and not be absolute value functions? Like an assymetrical v shape?
• Yes, piecewise functions have a different calculation that determines the shape of each piece.
• Can u flip it on the y axis as well?
• Yes you can flip it on y axis if the points are on the left or right side of the axis.
• Hi, is -7|x| equal to -|7x|? Can you write -7|x| as -|7x|?
• Yes, those are equivalent expressions.
• Is there any way to tilt or rotate the function? I.e rotate the function by say 45 degrees.
• I don't think so. Wouldn't the function then change, because of each input having only one output?
• how did he graph that y=-7 absolute value of x
• Read the problem at the top. If asks for the graph of y=|x| reflected across the x-axis and then scaled vertically by a factor of 7. It also asks for the equation for that graph.
• why does the 7 on the x axis start near 1
• Sal has 3 graphs:
y=|x|
y=-|x|
y=-7|x|

I also don't see any point on a graph that is (7,1). If x=7, here are the points that would be on each graph.
The 1st graph y=|x| would have a point (7,7)
The 2nd graph y=-|x| would have a point (7,-7)
The 3rd graph y=-7|x| would have a point (7,-49)

I don't know if this helps. If you clarify your question, may be I can help further.
• if it says scaled vertacally by factor of 7 how does -7 replace y
• The -7 doesn't replace y. The equation becomes: y=-7|x|.
The "-" does the reflection across the x-axis.
The 7 does the vertical scaling by a factor of 7.

Hope this helps.

## Video transcript

- [Instructor] The graph of y is equal to absolute value of x is reflected across the x-axis and then scaled vertically by a factor of seven. What is the equation of the new graph? So pause the video and see if you can figure that out. Alright, let's work through it together now. Now, you might not need to draw it visually but I will just so that we can all together visualize what is going on. So let's say that's my x-axis and that is my y-axis. y equals the absolute value of x. So for non-negative values of x, y is going to be equal to x. Absolute value of zero is zero. Absolute value of one is one. Absolute value of two is two. So it's gonna look like this. It's gonna have a slope of one and then for negative values, when you take the absolute value, you're gonna take the opposite. You're gonna get the positive. So it's gonna look like this. Let me see if I can draw that a little bit cleaner. This is a hand drawn sketch so bear with me but hopefully this is familiar. You've seen the graph of y is equal to absolute value of x before. Now, let's think about the different transformations. So first, they say is reflected across the x-axis. So for example, if I have some x value right over here, before, I would take the absolute value of x and I would end up there but now we wanna reflect across the x-axis so we wanna essentially get the negative of that value associated with that corresponding x and so for example, this x, before, we would get the absolute value of x but now we wanna flip across the x-axis and we wanna get the negative of it. So in general, what we are doing is we are getting the negative of the absolute value of x. In general, if you're flipping over the x-axis, you're getting the negative. You're scaling the expression or the function by a negative. So this is going to be y is equal to the negative of the absolute value of x. Once again, whatever absolute value of x was giving you before for given x, we now wanna get the negative of it. We now wanna get the negative of it. So that's what reflecting across the x-axis does for us but then they say scaled vertically by a factor of seven and the way I view that is if you're scaling it vertically by a factor of seven, whatever y value you got for given x, you now wanna get seven times the y value, seven times the y value for a given x. And so if you think about that algebraically, well, if I want seven times the y value, I'd have to multiply this thing by seven. So I would get y is equal to negative seven times the absolute value of x and that's essentially what they're asking, what is the equation of the new graph, and so that's what it would be. The negative flips us over the x-axis and then the seven scales vertically by a factor of seven but just to understand what this would look like, well, you multiply zero times seven, it doesn't change anything but whatever x this is, this was equal to negative x but now we're gonna get to negative seven x. So let's see, two, three, four, five, six, seven so it'd put it something around that. So our graph is now going to look, is now going to look like this. It's going to be stretched along the vertical axis. If we were scaling vertically by something that had an absolute value less than one then it would make the graph less tall. It would make it look, it would make it look wider. Let me make it at least look a little bit more symmetric. So it's gonna look something, something like that but the key issue and the reason why I'm drawing is so you can see that it looks like it's being scaled vertically. It's being stretched in the vertical direction by a factor of seven and the way we do that algebraically is we multiply by seven and the negative here is what flipped us over the x-axis.