Scaling & reflecting absolute value functions: graph

function G can be thought of as a stretched or compressed version of f of X is equal to the absolute value of X what is the equation for G of X so you can see f of X is equal to the absolute value of x here in blue and then G of X not only does it look stretched or compressed but it also is flipped over the x-axis so like always pause this video and see if you can come up yourself with the equation for G of X all right now let's work through this together so there's a couple of ways we could do it we could first try to flip f of X and then try to stretch or compress it or we could stretch or compress it first and then try to flip it well actually let's flip it first so let's say that we have a function that looks like this it's just exactly what f of X is but flipped over the x-axis so let's just flip over the x-axis so all the values for any given X whatever Y you used to get you're not getting the negative of that so this graph right over here this would be the graph I'll call this Y is equal to the negative absolute value of x whatever the absolute value of x would have gotten you before you're now going to get the negative or the opposite of it so this is getting us closer to our definition of G of X the key here is how do we appropriately stretch or squeeze this green function so let's think about what's happening on this green function when X is equal to 1 the function itself is equal to negative 1 but we want it if we want it to be the same as G we want it to be equal to negative 4 so it's actually 4 times the value for given X at least for x equals 1 G is giving me something 4 times the value that my green function is giving it's not just true for positive X's it's also true for negative X's you can see it right over here when X is equal to negative 1 my green function gives me negative 1 but G gives me negative 4 so it's giving me 4 times the value is going to be 4 times a negative value so it's going even more negative so what you can see to go from green 2g you have to multiply this thing right over here by four so that is what essentially stretches it down stretches it down in the vertical direction so we could say that G of X is equal to it's not negative absolute value of x negative four times the absolute value of x and you could have done it the other way you can say hey let's complete first stretch or compress F and say all right before we even flip it over here to unsleep key it would look like this if we were on flip G it would look it would look like it would look like this if we were to unsleep G so this thing right over here this thing looks like four times f of X we could write this as we could write this as Y is equal to four times f of X or you could say Y is equal to four times the absolute value of x and then we have a negative sign whatever you have whatever positive value are getting before you now get the opposite value and that would flip it over and get you G which is exactly what we already got