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### Course: Algebra 1>Unit 10

Lesson 1: 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.
(8 votes)
• 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)
(16 votes)
• what scaled even mean?
(7 votes)
• "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.
(12 votes)
• The 7 is the slope, right?
(5 votes)
• 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.
(15 votes)
• Can u flip it on the y axis as well?
(2 votes)
• Yes you can flip it on y axis if the points are on the left or right side of the axis.
(4 votes)
• Why do we call it "vertical" scaling? What's the difference between scaling such graphs vertically and horizontally?
(2 votes)
• Great question! 😊 When we talk about "vertical" scaling, we're talking about stretching or squishing the graph up or down. This happens when we multiply the whole function (like by a number.)

On the other hand, "horizontal" scaling means stretching or squishing the graph left or right. This happens when we multiply the input by a number.

So, in the video, when we multiply the absolute value function by -7, we're doing a vertical scale (and a reflection because of the negative sign)! - Khanmigo AI, (thought I'd try the handy ai feature!)
(2 votes)
• So scaling by a factor of 7 is different from just moving the function up by 7 like in the previous video?
(2 votes)
• Yes. Scaling is sort of like stretching the graph vertically, where as moving the function up preserves the shape. If the function is x, scaling it would produce 7x, where as moving it upwards produces x+7. If you graph both functions, you can clearly see the difference.
(1 vote)
• can functions have graphs in a v shape and not be absolute value functions? Like an assymetrical v shape?
(2 votes)
• Yes, piecewise functions have a different calculation that determines the shape of each piece.
(1 vote)
• Is there any way to tilt or rotate the function? I.e rotate the function by say 45 degrees.
(1 vote)
• I don't think so. Wouldn't the function then change, because of each input having only one output?
(2 votes)
• Confused. Since x is absolute value it is always positive? How can the graph go to the left of zero?
(1 vote)
• The value of "x" doesn't need to be positive. The absolute value turns "x" into a positive. For example, if x=-3, then |-3|=+3. The absolute value is impacting the value of Y, not X.

Hope this helps.
(2 votes)
• What if we scaled horizontally by a factor of 7?
(1 vote)
• Scaling horizontally by a factor of 7 means that instead of 𝑥 we get 7𝑥.
So, scaling 𝑦 = −|𝑥| horizontally gives us 𝑦 = −|7𝑥|.
(2 votes)

## 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.