Chapter 7B Methods unit 1&2 pls help

[Pearsoncomeatme]

Can someone please explain dilations? What does (x,y) into (x,by) even mean? How would I dilate -2,-3 by a factor of 4 in the x-axis and y-axis. I genuinely don't know anything, and I'm not even doing methods, like bruh.
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If someone could please explain im not even joking I will worship u

[The Cat In The Hat]

Can someone please explain dilations? What does (x,y) into (x,by) even mean? How would I dilate -2,-3 by a factor of 4 in the x-axis and y-axis. I genuinely don't know anything, and I'm not even doing methods, like bruh.
]
If someone could please explain im not even joking I will worship u

I had problems with this myself. (x, y) into (x, by) means that it has been changed so that the x-value stays the same, however it has been dilated by a factor of b in x-axis. To figure this out when I was confused I used that a factor of b from the x-axis meant it was moving AWAY from the x-axis, which therefore changed it in the y-axis. And therefore dilating by a factor of b from the y-axis means moving away from the y-axis, so changing in the x-axis ('away' meaning away or towards).

To dilate (-2, -3) by a factor of 4 in the x-axis - the x-axis means you're changing the y value, so that would be (-2, 4*-3), which is (-2, -12).

To dilate (-2, -3) by a factor of 4 in the y-axis - the y-axis changes the x value, becoming (4*-2, -3) which is (-8, -3).

To dilate by a factor of 4 in the x-axis and a factor of 4 in the y-axis, do both of these steps, resulting in (4*-2, 4*-3) which is (-8, -12).

Someone else please check my explanation, I only do this off my memory...

I feel this isn't a very clear explanation. If you want more help, ask. Hope it helps though.

[Pearsoncomeatme]

Yes, I got the same answers honestly, but I still really can't wrap my head around it for some reason. Idk if you have the textbook but in the chapter 7b it says:

Section Summary:

Applying the dilation from the x-axis (x,y) > (x,by) (which i understand i guess?)
Replacing y with y/b in the equation to obtain y=bf(x)


LIKE WHAT DOES THAT MEAN? why is there so many letters, i really have no idea why thats included, is it important? I have no idea

[Pearsoncomeatme]

I had problems with this myself. (x, y) into (x, by) means that it has been changed so that the x-value stays the same, however it has been dilated by a factor of b in x-axis. To figure this out when I was confused I used that a factor of b from the x-axis meant it was moving AWAY from the x-axis, which therefore changed it in the y-axis. And therefore dilating by a factor of b from the y-axis means moving away from the y-axis, so changing in the x-axis ('away' meaning away or towards).

To dilate (-2, -3) by a factor of 4 in the x-axis - the x-axis means you're changing the y value, so that would be (-2, 4*-3), which is (-2, -12).

To dilate (-2, -3) by a factor of 4 in the y-axis - the y-axis changes the x value, becoming (4*-2, -3) which is (-8, -3).

To dilate by a factor of 4 in the x-axis and a factor of 4 in the y-axis, do both of these steps, resulting in (4*-2, 4*-3) which is (-8, -12).

Someone else please check my explanation, I only do this off my memory...

I feel this isn't a very clear explanation. If you want more help, ask. Hope it helps though.

But also thank you so much tho   

[keltingmeith]

Yes, I got the same answers honestly, but I still really can't wrap my head around it for some reason. Idk if you have the textbook but in the chapter 7b it says:

Section Summary:

Applying the dilation from the x-axis (x,y) > (x,by) (which i understand i guess?)
Replacing y with y/b in the equation to obtain y=bf(x)


LIKE WHAT DOES THAT MEAN? why is there so many letters, i really have no idea why thats included, is it important? I have no idea

Maybe some pictures and ACTUAL numbers will help:

Let's say I want to do a dilation. All of the following statements are EXACTLY the same:

Dilate by a factor of 3 from the x-axis
Map (x, y) onto (x, 3y)
Graph the equation y=3*f(x)

And all of them look like this. Notice that the original function, f(x), is drawn in green. The graph which has been dilated by a factor of 3 from the x-axis is drawing in blue. Notice how the graph gets steeper/bigger/further away from the x-axis, EXCEPT when the graph intercepts the x-axis? Well, you can think of dilating like this: let's say at ever x-intercept, you nail the graph to the ground. Next, you tie a rope to the graph, and start pulling - and you pull, and pull, and stretch this graph, until the graph is 3x BIGGER than it was before. This is what a dilation means. This is what it graphically looks like. (I've also included dilation by a factor of a 1/3 in red, for interest. In this case, you can use the same analogy - but instead of pulling AWAY from the x-axis, you pull TOWARDS it instead)

Okay, that's all neat and swell - what about for other numbers? Maybe instead I'm interested in the transformation:

Map (x, y) onto (x, 2y)
Dilate by a factor of 2 from the x-axis

Well, you can see those transformation by clicking on the circles on the side. Notice that the graph for y=2f(x) (which should be in purple) is half-way between the blue and green graphs? That's because you've no longer pulled the graph so that it's 3x bigger - now you've only pulled it so it's 2x bigger. You can see something similar if you click the circle next to the y=f(x)/2 equation - this should show an orange graph, and it's not AS small as for the y=f(x)/3 equation.

There's one final thing you can do here - I've included a slider, and an equation for y=b*f(x). Click the circle to display y=b*f(x) (it should show up in black), and then play with the slider for b. As you can see, if b is a bigger number, the graph gets bigger - and if b is a smaller number, it gets smaller.


So maybe you already understood this, maybe it already made sense for graphs - so what about for points, as in your first question? Well, a graph is REALLY just a set of points! For example, the graph of y=x^2 is just the points:

(1, (1)^2), (2, (2)^2), (3, (3)^2), (4, (4)^2), and so on and so on

So, if you were asked to dilate the graph of y=x^2 by a factor of 2 from the x-axis, you know that this would turn the equation into y=2x^2, and so the points would know be:

(1, 2*(1)^2), (2, 2*(2)^2), (3, 2*(3)^2), (4, 2*(4)^2), and so on

So then - how would you do this for just a point? Well, I came up with these numbers by feeding them into a new equation - but can you see that I've also made them by multiplying the original y-value by 2? I can do the exact same thing for ANY point, even if I don't know what equation it came from. So let's say I have a point, (2, 3). I don't know what equation made that point - but I want to dilate by a factor of 2 from the x-axis. In that case, the point would NOW be (2, 6).


Golden rule of mathematics - yes, letters are scary. But they're there to help you, so you kinda have to get used to them. One way to help you get used to them - just replace them with numbers. So, if seeing that b EVERYWHERE is scaring you, you should look at what happens when you replace it with 3. If you replace b with 3 EVERYWHERE you see it, then do some graphing, maybe it makes sense. If it doesn't, then try it again with 2. Then try it again with 5. Then maybe use desmos to make a slider (like I did). And maybe after all that, it'll make sense what b means.

Hopefully that's explained everything a bit for you.

[The Cat In The Hat]

So normally y = f(x). When y becomes y/b, rearrange to y = b * f(x). I assume you understand what that means?

To dilate (x, y) into (x, by): if y = f(x), the dilation becomes y = bf(x), which could be shown as y/b. I won't try and explain further because I've always had trouble with this one myself. I found just straight tons of practise helped, practise where you do it, then check the answer, if it's wrong, do it again until you get it right - use worked solutions if really stuck (I never got to that point quite).

Hope this helps. But I say again, practice is the important thing here. Take it as a rule that you must abide by, even if it doesn't make sense, use it, and gradually it becomes clearer and makes more sense. Or, that's what I found anyway.

[Pearsoncomeatme]

Thank you so much, I went through 7c, and it was a lot easier after your explanations. I think the function notation threw me off a lil at the start, but I got it