SM2 8.2-1: Graphing Horizontal Dilations and Reflections

Ajouté : 2026-06-02

[00:00:00]Hello and welcome. So we're going to graph some  transformations today. Let's get started. First thing you need to figure out is what are you  looking at. Anything that's going to have a parabolic shape, that's going to be an x  squared, okay? Anything that's going to be, I call it a half seagull, is going to be a square  root. Anything that's going to be a V shape is your absolute value, okay? Now keep in mind that  for all three of these graphs in order to graph it correctly your turning point, your vertex, must  be the first point that you plot. Otherwise it won't plot it correctly. So keep that in mind.  Alrighty. So here we have a square root, right?

[00:00:39]We see that radical symbol which means you're  going to have that half seagull. Now for those half seagulls I'd say okay the normal points that  we plot for these ones it always starts at 0, 0.

[00:00:51]So if you plug in 0, square root of 0 is  0, right? Your next point is over 1 up 1 because if you plug in 1, square root of 1 is  1. Your next nice point is going to be at 4, 2.

[00:01:08]We'll call it a nice point because it's a  perfect square. What's the square root of 4?

[00:01:13]2. And that's where that point came from, okay?  So your pattern is 0, 0 over 1 up 1, over 4 up 2, and then it'll keep going from there, okay? So  now let's figure out what our transformation is. And I'd say okay. For our transformation we  see that you have a negative one-fifth being multiplied to x. Now multiplication implies  that you have some kind of dilation going on, all right? Now whether that's horizontal or  vertical you have to look at where the number is. Because the negative one-fifth  is inside of the radical symbol, inside of your operation, that means  that you are looking at a horizontal dilation specifically, okay? Now this  negative out front implies a reflection.

[00:02:07]So we have technically two different  transformations happening here. We have a reflection as well as a horizontal dilation.  Now that dilation can mean either stretch or compression, or shrink, if you want to call it  shrink instead of compression, okay? Either or.

[00:02:22]And you'd say all right, well, now let's figure that out.  Stretching or shrinking. Now keep in mind that you always do the reciprocal of what is inside with x  with your dilations. So if it's negative one-fifth the reciprocal of that would be negative five  because that's technically negative five over one. You just flip the fraction. Reciprocals  don't change signs. So if it's negative before, negative after. Now the reason why we use the  reciprocal, think about it this way. If you have some absolute value of x equals 10 that means  that you would just plug in 10. Absolute value of 10 is 10. So next you'd say okay what if it's one  half x horizontal dilation? What would that mean for x specifically? Well that means you'd have to  plug in 20 for x. x would have to be twice as much as it was before in order to get 10 back  because half of 20 is 10. It's the only way to get back to 10. And the next one  what if it was 2x? Well in order to get to 10 you don't need as much. You only need  specifically you only need half as much, okay? Because 2 times 5 is 10. That's how you get  back to 10 on that one. So you started with 10, right? In order to get back to 10 you'll need  half as much when there's a 2 in front. So in other words, reciprocal because the reciprocal  of one-half is two. Reciprocal of two, one-half.

[00:03:54]Alrighty. So to graph this once we know our  pattern that it starts at 0, 0. You can't change or multiply zero, zero, and it hasn't  changed places. It's still at zero, zero, okay? And now your next point is over 1 up 1  but we know we're not going over 1. We know we're going over 5. Now what direction we talk  about when we're talking about over is dependent on that reflection. Left is the negative. So it's  going to be not left not-- heh golly, not right one up one it's going to be left 5 up 1. And the reason why the y value  hasn't changed, why all three of these will remain the same, is because this dilation was  only on the inside. It only affected the x's, okay? Which means that our next one is supposed to  be over 4 right 4 but it's not going to the right.

[00:04:52]The negative says we're going to the left and then  4 times 5 that's negative 20. That's way beyond our graph. We can't even put that second point  on there, okay? Oh that was like a wonky shape.

[00:05:12]Not the best but it'll do.

[00:05:19]Okay, assume that these lines are going where  I want them to go. But when it comes to these that that would be it. That would be your  graph. You plot this point first then this one and you would be done, okay? So to keep in  mind that when you have horizontal it's going to do the reciprocal with stretches and  shrinks. All right. Thanks for watching.