in this lesson you're going to learn how to graph absolute value functions with transformations we're going to go through three examples so let's dive in the first thing you want to know is the basic shape of the parent function this y equals absolute value of x graph and the way you would get points for this graph is by picking a few negative zero and a few positives so let's go ahead and do that so negative three when we put it into between these absolute value bars that tells us to make the quantity positive it's like the distance from zero and so this is going to be three if we put negative two in and we get the absolute value that gives us positive two negative one is one zero is zero one is one it's always going to be positive remember it's that distance from zero and so on so if we plot these points we're going to get negative 3 3 negative 2 2 negative 1 1 0 0 1 1 2 2 3 3 and it keeps going like that and notice how we're getting this you know real sharp v this real sharp uh corner here as opposed to a quadratic where it's got that u shape that parabola shape so this point here where the graph bends this is referred to as the vertex and we're going to be looking at that closer when we graph absolute value graphs that are in this form y equals a absolute value of x minus h plus k the h and the k is involved with shifting the graph left and right up and down and the a is involved with stretching the graph so let's talk a little bit more about that so you see this h here that's grouped with the x this is going to shift it in the x axis direction like left and right but you want to remember that this quantity here has actually the opposite effect on the graph what i mean by that is if you had x minus one it looks like it's minus one you'd think it would go left one it actually goes positive one it goes the opposite way whereas over here this k value this is what's involved with shifting the graph up and down if this is plus two it would actually go up two if it's minus two it'd go down two say for example this was uh absolute value of x plus three see plus three you would think would be right three right but it's actually the opposite it goes to the left three now the a value if it's greater than one it's going to be a vertical stretch meaning that the graph is going to be narrower it's like you're pulling it in the vertical direction if a is between zero and one we call it a vertical shrink or a vertical compress it's going to make it wider like that and if it's negative it's going to reflect it over the x-axis meaning it's going to open down like that so let's jump into the three examples and let's practice so for number one we have y equals the absolute value of x plus one minus three so the first thing i like to do is i like to find that vertex that point where the graph bends and this is going to shift left one and down three so left one down three so you're going to be right there that's your vertex and then the a value here you don't actually see an a value which means that this is really like one because one times anything is itself so we usually don't write the one but you can think of it as a one here in front and one is like one over one right anything divided by one is itself and the reason i do that is because you can think of this a value like the slope of a line so if the slope is 1 that means from this point we're going to go rise one run one and because this absolute value graph is symmetric about this axis of symmetry if you fold it it's going to map to itself because we went up one and right one we can also go up one and left one it's like i'm folding it over the vertex like that and we can continue we can go up one over one and same thing here up one over one and up one over one and up one over one etcetera and so you can see you're going to get that real nice v shape absolute value graph now if you're interested in finding the domain and range the domain is what the x values can be and you can see that this graph is going to the left and the right forever and ever so the domain would be all real numbers but the range those are what the y values can be and here you can see that the lowest y can be is negative 3 or greater so for the range we'd say y is greater than or equal to negative three okay let's do another example see if you can do this one number two it says y equals negative two times the absolute value of x minus three plus four so where is that vertex well remember this one here in the parentheses again the absolute value bars i should say has the opposite effect the minus 3 is actually going to shift it right 3 and the plus 4 is going to have the same effect as that positive sign it's going to go up positive 4. so we're going to go right 3 up 4 1 2 3 4 right about there that's our vertex but this a value is negative so we know that the graph is going to be opening down something like that and the negative 2 we can think of like the slope that's like negative two over one so from here i'm going to go down two one two and right one and i'm gonna reflect it so i could go down two and left one and i'll repeat that again down to left one from here down to uh right one so you see how that slope but i'm then reflecting it over this axis of symmetry like that so here on this one again the domain is all real numbers the range is going to be y is less than or equal to four see four is the highest or below so we'll just write that down positive four okay one more example if you're enjoying this video so far be sure to check out my algebra one and my algebra 2 video courses as well as my huge act math review video course and my huge sat math review video course i'll put a link in the description below but number three what we have is y equals one third absolute value of x plus one minus two so what do you think for this one where is the vertex or you could think about it as you know what's the transformation where's where's the graph being shifted or stretched we can see this one that's grouped with the x we talked about how that shifts it in the x direction but it has the opposite effect so plus one is actually shifting at left one this one here has the same effect it's going to go minus 2 that's down 2. so that's going to put us right there that's our vertex and we can treat the a value like the slope right like the slope of a line rise over run we know it's positive so the graph is going to open up like that the one-third though is between zero and one so this is going to be like a vertical shrink or a vertical compression it's going to make the graph wider but let's go ahead and graph it so from the vertex i'm going to go up one that's the rise run three one two three i can repeat that up one and then three to the right i could also go the other direction rise one and go left three same thing again rise one left three and that's because it's symmetric about the axis of symmetry so the graph is going to look something like that the domain is all real numbers and the range is going to be y is greater than or equal to negative 2 and you got it if you want to see another example follow me over to that video right there and i'll show you some more practice problems