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Current time:0:00Total duration:6:03

CCSS.Math:

so this is a screenshot of desmos it's an online graphing calculator and what we're going to do is use it to understand how we can go about scaling functions and I encourage you to go to desmos and try it on your own either during this video or after so let's start with a nice interesting function let's say f of X is equal to the absolute value of x so that's pretty straightforward now let's try to create a scaled version of f of X so we could say G of X is equal to well I'll start with just absolute value of x so it's the same as f of X so it just trace to G of X right on top of F but now let's multiply it by some constant let's multiply it by 2 so notice the difference between G of X and f of X and you can see that G of X is just 2 times f of X in fact we can write it this way we can write G of X is equal to 2 times f of X we get to the exact same place but you can see that as as our x increases G of X increases twice as fast at least for positive X's on the right-hand side and actually as X decreases G of X also increases twice as fast so it's not just a coincidence that we have a 2 here and it increased twice as fast well let's put a 3 here well now it looks like it's increasing 3 times as fast and it does that in both directions now what if we were to put a zero point five year zero point five well now it looks like it's increasing half as fast and that makes sense because we are just multiplying we are scaling how much our f of X is so before when x equals one we got to one but now when x equals 1 we can only get to one half before when x equals five we got to 5 now when we go to x equals five we only get to two point five so we're increasing half as fast or we have half the slope now an interesting question to think about is what would happen if instead of it just being an absolute value of x let's say we were to have a non zero y-intercept so let's say I don't know plus 6 so notice then when we change this constant out front it not only just changes the slope but it changes the y-intercept because we're multiplying this entire expression by 0.5 so if you multiply it by 1 we're back to where we got before and now if we multiply it by 2 this should increase the y-intercept because remember we're multiplying both of these terms by 2 and we see that it not only doubles the slope but it also increases the y-intercept if we go to 0.5 not only did it decrease the slope by a factor of 1/2 or or I guess you could say multiply the slope by 1/2 but it also made our y-intercept be half of what it was before and we can see this more generally if we just put a general constant here and we can add a slider and actually let me make the constant go from 0 to 10 with a step of 0.05 that's just how much does it increase every time you change the slider and notice when we increase our constant not only we're getting narrower because the slope the magnitude of the slope is being scaled but our y-intercept increases and then as K decreases our y-intercept is being scaled down and our slope is being scaled down now that's one way that we could go about scaling but what if instead of multiplying our entire function by some constant we instead just replace the X with a constant times X so instead of K times f of X what if we did f of K times X another way to think about is G of X is now equal to the absolute value of K X plus 6 what do you think is going to happen pause this we didn't think about it well now when we increase K notice it has no impact on our y-intercept because it's not scaling the y-intercept but it does have an impact on slope when K goes from 1 to 2 once again we are now increasing twice as fast and then when K goes from one to one half we're now increasing half as fast now this is with an absolute value function what if we did it with a different type of function let's say we did it with a quadratic so 2 minus X the squared and let me scroll down a little bit and so you can see when K equals 1 these are the same and now if we increase our K let's say we increase our K - 2 notice our parabola is our in this case decreasing as we get further and further from zero at a faster and faster rate that's because what you would have seen at x equals 2 you're now seeing at x equals 1 because you are multiplying 2 times that and so and if we in and then if we go between 0 and 1 notice on either side of 0 our parabola is decreasing at a lower rate it's a changing rate but it's a lower changing rate I guess you could put it that way and we could also try just to see what happens with our parabola here if instead of doing K X we once again put the K out front what is that going to do and notice that is changing not only how fast the curve changes at different points but it's now also changing the y-intercept because we are now scaling that y-intercept so I'll leave you there this is just the beginning of thinking about scaling I really want you to build an intuitive sense of what is going on here really think about mathematically why it makes sense and go onto desmos and play around it with yourself and and also try other types of functions and see what happens