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# Graphing absolute value functions

CCSS.Math:

## Video transcript

so we're asked to graph f of X is equal to 2 times the absolute value of x plus 3 plus 2 and what they've already graphed for us this right over here this is the graph of y is equal to the absolute value of x so let's do this through a series of transformations so the next thing I want to graph let's see if we can graph y y is equal to the absolute value of x plus 3 now in previous videos we have talked about it if you replace your X with an X plus 3 this is going to shift your graph to the left by 3 you could view this as the same thing as y is equal to the absolute value of x minus negative 3 and whatever you are subtracting from this X that is how much you are shifting it's you're going to shift it 3 to the left and we're going to do that right now and then we're just going to confirm that it matches up that it makes sense so let's first graph that you get my ruler tool here so if we shift three to the left it's going to look something like it's going to look something like this so on that when whatever we have inside the absolute value sign is positive we're going to get essentially this slope of 1 and whenever we have inside the absolute value sign is negative we're going to have a slope of essentially negative 1 so it's going to look it's going to look like that and let's confirm whether this actually makes sense below x equals negative 3 for X values less than negative 3 what we're going to have here is this inside of the absolute value sign is going to be negative and so then we're going to take its opposite value and so that makes sense that's why you have this downward line right over here now for X is greater than negative 3 when you add 3 to it you're going to get a positive value and so that's why you have this upward sloping line right over here and at x equals negative 3 you have 0 inside the absolute value sign just is if you didn't shift it you would have had zero inside the absolute value sign at x equals zero so hopefully that makes a little bit more intuitive sense of why if you replace X if you replace X with X plus three and this isn't just true of absolute value functions this is true of any function if you replace X with X plus three you're going to shift three to the left all right now let's keep building now let's see if we can graph y is equal to two times the absolute value of x plus three so what this is essentially going to do is multiply the slopes by a factor of two it's going to stretch it vertically by a factor of two so for X values greater than negative three instead of having a slope of one you're going to have a slope of two so let me get my ruler out again and see if I can draw that so we put that there and then so here sort of a slope of one I'm going to have a slope of two let me draw that it's going to look like that right over there and then instead of having a slope of negative 1 for values less than x equals negative three I'm going to have a slope of negative 2 let me draw that right over there so that is the graph of y is equal to two times the absolute value of x plus three and now to get to the f of X that we care about we now need to add this two so now I want to graph and we're in the homestretch I want to graph y is equal to two times the absolute value of x plus three plus two well whatever Y value I was getting for this orange function I now want to add two to it so this is just going to shift it up vertically by two so instead of so this is going to be shifted up by two this is going to be shifted that every point is going to be shifted up by two or you can think about shifting up the entire graph by two here and the orange function our y-value I got for the black function I'm going to have to get to more than that and so it's going to look it's going to look like this so let me see I'm shifting it up by two so for X less than negative three it'll look like that and for X greater than negative three it is I'm going to look like it is going to look like that and there you have it this is the graph of y or f of X is equal to two times the absolute value of x plus three plus two and you could have done it in different ways you could have shifted up to first then you could have multiplied by a factor of two and then you could have shifted and then so you could have moved up to first then you could have multiplied by a factor two then you could have shifted left by three or you could have multiplied by a factor of two first shift it up two and then shift it over so there's multiple there's three transformations going up here this is an this is a let me color them all so this right over here tells me this over here says hey shift left shift left by three this told us stretch vertically by two or essentially multiply the slope by two stretch vert by two and then that last piece says shift up by two shift up by two which gives us our final result for f of X