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Current time:0:00Total duration:8:09

Scaling functions horizontally: examples

CCSS.Math:

Video transcript

we are told this is the graph of function f fair enough function G is defined as G of X is equal to F of 2x what is the graph of G so pause this video and try to figure that out on your own alright now let's work through this and the way I will think about it I'll set up a little table here and I'll have an X column and then I will have and then I'll have a well actually just input G of X column and of course G of X is equal to F of 2x so when X is and actually let me see when X is equal to I could pick a point like X equaling 0 so G of 0 is going to be f of 2 times 0 so it's going to be F of 2 times 0 which is still F of 0 which is going to be equal to a little bit over 4 so which is equal to F of 0 and so they're going to both have the same y-intercept but interesting things are going to happen the further that we get from the y-axis or as our x increases in either direction away or as X gets bigger at either direction from 0 so let's think about what's going to happen at x equals 2 so at x equals 2 G of 2 is going to be equal to F of 2 times 2 2 times 2 which is equal to f of 4 and we know what F of 4 is F of 4 is equal to 0 so G of 2 is equal to F of 4 which is equal to 0 so notice the corresponding point is kind of gotten compressed in or squeezed in or squished in in the horizontal direction and so what you see happening at least on this side of the graph is everything's happening a little bit faster your your whatever was happening at a certain X it's now happening at half of that X so this side of the graph is going to look something dried out a little bit better than that it's going to look something like like this like this everything's happening twice as fast and what happens when you go in the negative direction well think about what G of negative 2 is G of negative 2 is equal to F of 2 times negative 2 2 times negative 2 which is equal to F of negative 4 which we see is also equal to 0 so G of negative 2 is 0 and you might be thinking why don't you pick 2 a negative 2 well the intuition is that things are going to be squeezed in and things are happening twice as fast so whatever was happening at x equals 4 is not going to happen at x equals 2 whatever is happening at x equals negative 4 is now we're going happen at x equals negative 2 and I saw that we were at very clear points at x equals negative 4 and x equals 4 on f so I just took half of that to pick my X values right over here and then so what our graph is going to look like is something like this it's going to look something like this it's going to look like it's been squished in from the right and the left now let's do another example so now they've not only given the graph of F they've given an expression for it what is the graph of G of X which is equal to this business so pause this video and try to figure that out all right the key is to figure out the relationship between f of X and G of X and what we can see the main difference is is instead of an X here in f of X we have an x over 2 so everywhere there was an X we've been replaced with an x over 2 so another way of thinking about it is G of X is equal to F of not X but f of X over 2 or another way of thinking about it G of X is equal to F of 1/2 X and then we can do a similar type of exercise and they've given us some interesting points the points to the point or the point x equals 2 the point x equals 4 and the point x equals 6 so let's think about this last time when it was G of X is equal to 2x things were happening twice as fast now things are going to happen half as fast and so what I would do let me just set up a little table here the interesting x-values for me are the ones that if I take half of them then I'm going to get one of these points so actually let me write this 1/2 1/2 X and then I can think about what G of X is equal to f of one half X is going to be so I want my one half X to be let's see it could be two four and six two four and six and why did I pick those again well it's very clear what values f takes on at those points and so if one-half X is 2 then X is equal to 4 if one-half X is 4 then X is equal to 8 if X is equal to 12 then 1/2 X is 6 and so then we could say all right G of 4 is equal to F of 2 which is equal to 0 that's why I picked 2 4 & 6 it's very easy to evaluate F of 2 F of 4 and F of 6 it gave us those points very clearly so G of 8 is going to be equal to G is going to be equal to F of 1/2 of 8 or F of 4 which is equal to negative 4 and then G of 12 is equal to F of 6 which is half of 12 which is equal to 0 again so then we can plot these points if we get a general sense of the shape of the graph so let's see G of 4 is equal to 0 G of 8 is equal to negative 4 all right over there and then G of 12 is equal to 0 again so everything has been stretched out so there you go it's been stretched out and at least in the horizontal Direction is one way to think about it in the horizontal direction and you can see that this point in F corresponds to this point in G it's gotten twice as far from the origin because everything is growing half as fast you input a next you take 1/2 of it and then you input it into F and then this point right over here corresponds to this points that are happening at for this vertex point it's now happening in 8th and last but not least this point right over here corresponds to this point instead of happening at 6 it's happening at 12 everything is getting stretched out let's do one more example f of X is equal to all of this we have to be careful there's a cube root over here and G is a horizontally scaled version of F the functions are graphed where F is solid G is dashed what is the equation of G so pause this video and see if you can figure that out alright let's do this together and it looks like they've given us some points that seem to correspond with each other to go from F to G it looks like these corresponding points have been squeezed in closer to the origin and what we can see is is that F of negative 3 F of negative 3 seems to be equal to G of negative 1 and F of 6 over here F of 6 seems to be equal to G of 2 G of 2 or another way to think about it whatever X you input in G it looks like that's going to be equivalent to 3 times that X input it into F so G of X is equal to F of 3 X and so if you want to know the equation of G we just evaluate F of 3 X so f of 3 X is going to be equal to and I could just actually put an equal sign like this F of 3 X is going to be equal to negative 3 times the cube root of instead of an X I'll put a 3x right over there 3 X plus 2 and then we have plus 1 and that's it that's what G of X is equal to it's equal to F of 3 X which is that we substituted this X with a 3x and we are done