SM2 8.2-4: Writing the Equation of a Graph With a Vertical Dilation and/or Reflection

Added: 2026-06-02

[00:00:00]Hello and welcome. So we're going to be  writing a equation given a graph. So first thing you got to know is you got to  know what your parent functions look like, right? There are three main ones that we're  looking at. So first one is a parabola, right?

[00:00:15]Parabolic in shape where you'd say okay that is  formed from a power of two, exponent of two. Next one you have a half seagull looking graph where  it's going to be square root of x. And then your last one is a v-shaped graph which is a result  from an absolute value of x. So if you look at these three shapes it's not a U it's not a V which  means this thing has to be a square root graph, okay? So we know in our final answer it's going to  have some kind of square root there. All right. So now we got to figure out okay what transformations  are happening. For a square root you have to know where it starts. It always starts at 0, 0, okay? We  know it's zero, zero. You plug in x you get out y, right? Zero-- square root of zero, zero, right?  If you plug in 1 what's the square root of 1?

[00:01:05]1, okay? The next nice point will be 4, 2 because  you take the square root of 4. Square root of 4 is 2, okay? So those are your points that you're  looking at. You said all right so now all we got to do from here is you got to figure out okay  have we shifted at all? No you're still at zero zero. That one is the same. Which means the only  transformations that have happened here is some kind of stretch, or compression, right? Depending  on how we want to write it. I would just recommend going with vertical compressions, okay? So I look  at this one and I'd say okay, so normally if we're looking at just vertical then I'd say okay, the  next one is supposed to be going over 1. Well hold on over 1 is positive 1 is to the right but  we didn't go to the right we went to the left.

[00:02:01]So what does that mean? We're multiplying our x  values by a negative 1, right? You have reflected it. So essentially you're going to have negative  x on the inside because of that reflection, okay? If your graph was going this direction  that means that you wouldn't have a reflection and it would just be a positive x in there. All  right. Okay so I said okay well we're going to the left. So left 1 and then we need to figure  out where this is down wise. Now we don't have a scale for where this point is. It looks like  one half but I don't know. It doesn't quite help us there. So we look at the next nice point. What  can we for sure say we have? And that's negative 4, at x equals negative 4 and then negative  1, okay? We know that normally if we're only changing the y values we know that  if you go over 4 you should be going up 2 but are we going up 2 here? No we're not. We know  we're going down, so it's definitely going to be a negative on the y's, okay? And in addition to  going down we're not going down as far, right?

[00:03:19]Instead of going to 2 we're at 1. So that means  that 2 got multiplied by something to give us 1.

[00:03:27]The only option here is one half, because  if you take half of 2 we get 1 back. So that means that we are shrinking  the graph. You have a compression where you're going over 1 and this it looked  like a half and it truly is one half there, okay? And then you go over 4 and instead of  down one-- sorry, down 2 you're going down 1. That's half as much.

[00:03:54]And this is it. This actually is the correct  equation for this graph where instead of going, you know, the normal up we have gone down which means  reflection, okay? And additionally we're going the other direction as well which means we have double  reflection. All right. And then the one-half essentially just makes it a vertical compression,  where it's kind of squishing it vertically, okay? Now is this the only way to write your  equation? No. You also could have seen this as a horizontal compression or sorry a horizontal  stretch, okay? But that would have been giving you the exact same answer as you would writing it  for vertical. So I usually focus on the vertical.

[00:04:43]It's a little easier to think about. But that  would be all for this one. Thanks for watching.