in this lesson we're going to talk about how to graph absolute value functions using transformations so this is going to be the parent function of the absolute value of x so basically it's a graph that looks like a v pointing in the upward direction now if you want to plot it using points here's what you need to do the center is zero so you want to choose two points to the right and two points to the left and then plug in the numbers to find the y value the absolute value of zero is zero the absolute value of one is the same thing it's one the absolute value of negative one is positive one and the absolute value of negative two is positive soon and then you can plot the points if you do that you're going to get a graph that looks like this now what about the absolute value of negative x let's say if the negatives on the outside how would the graph look like if the negative is on the outside it's going to reflect over the x-axis so instead of opening in the upward direction it's going to open downward now what if the negative is on the inside if it's on the inside doesn't matter it's still going to open upward the absolute value of any negative number will produce a positive result for example if you plug in positive 1 absolute value of negative one will still be positive one so the negative on the inside will not cause the graph to open downward but the negative on the outside will now what are the domain and range of these two functions the domain represent all of the possible x values as you can see x can be anything the lowest x value all the way to the left is negative infinity and the highest is infinity so it's all real numbers for the domain you can plug in any value into x now what about the range the domain for the other one is the same by the way it's negative infinity to infinity that's going to be true for all absolute value functions if there's no fractions no radicals no logarithmic equations it's going to be all numbers now as for the range what's the lowest y value that we see in the graph on the left notice that it goes down towards negative infinity and the highest y value is zero so the range is from negative infinity to zero now for the graph on the right the lowest y value is zero the highest is infinity so the range is from zero to infinity and zero is included so we need to use a bracket instead of a parenthesis now what about graphing an equation that looks like this this graph is going to shift two units to the left and so it's going to open upward because it's a positive outside of the absolute value function now if you want to plot points need to realize that the slope is 1. so once you plot the vertex as you travel one unit to the right go up one unit so that will take you to the point negative one one and as you travel one unit to left go up one because the slope is one on the right side the slope is positive one but on the left side the slope is negative one and then to find the next point travel one to the right and up one and the left side is just going to be reflection of the right side so the graph is going to look like this if there was a 2 in front the slope would be 2 it would the graph would be like more steeper so to speak it would be narrow and not as wide now what about the graph for this one x minus three this one is going to shift three units to the right and it's still going to open in the upward direction with a slope of one now what about the absolute value of x plus two and also negative absolute value of x minus 3. what's going to happen to the graph in the first example it's going to shift up 2 units and because we have a plus sign in front it's going to open in the upward direction for the second example it's going to shift down three units that's a vertical shift left and right represents a horizontal shift and it's going to open downward due to the negative sign now what are the range for the graph for each of these let's start with the first one what's the range notice that the lowest y value is 2 the highest is infinity so the range is going to be from 2 to infinity and 2 is included now for the one on the right the lowest y value is negative infinity the highest is negative three and negative three is included so the range is from negative infinity to negative three you start from the low y value and you stop at the highest y value and there's no breaks in between so what if we have a combination of transformations let's say this graph so it's going to shift two units to the right and up three units so here's the vertex and it's going to open upward with a slope of one to the right and with a slope of negative one to the left by the way for those of you who prefer to use tables here's what you want to do if you set the inside part equal to zero this will give you the x coordinate of the vertex which is two after that you want to make that your center point choose two points to the right of two and two points to the left and find the y values at those points when you plug in two y is going to be three when you plug in three the absolute value of three minus two is one plus three that's going to be four if you plug in one it's gonna be the same thing if you plug in four four minus two is two plus three is five if you plug in 0 you're going to get the same thing and once you plot it you're going to get a graph that looks like this the domain for that graph is all real numbers but the range the lowest y value is 3 the highest is infinity so the range is going to be 3 to infinity try this one 4 minus the absolute value of x plus 1. so the graph is going to shift one unit to the left and it's going to shift up four units so the vertex is at the point negative one comma four and the slope the number in front of the absolute value function is one and because there's a negative sign in front of it we know it's going to open in a downward direction but let's plot points though to get the next point as you travel one unit to the right you need to go down one same thing one to the left go down one and then one to the right go down one again and just repeat the process and so this is going to be a rough sketch of the graph and i missed all the points there so that's how you can graph it if you want to draw a more accurate sketch using points and as we can see the range is from negative infinity to four the highest y value is four and don't forget to put brackets because it includes four let's try one more example so let's graph this one so this time the slope is going to be 2. the graph is going to shift one unit to the right and it's going to shift up three units so it starts at the point one three as we travel one to the right we need to go up two units on the y axis so the next point is over here one to the left we need to go up two units and as we travel uh one more unit to the right and one to the left we need to go up two more units but i'm out of space so i'm just gonna have to go with this because i'm out of space let's do one more example so let's say if we have five minus three x minus one so i'm going to put this over here and draw a bigger graph this time so as we travel one unit to the right we need to go up five units based on this it shifts one to the right horizontally and vertically up five units so the first point is that one comma five now we have a slope of three and because it's negative we know it's going to open downward so as we travel one unit to the right needs to go down three units five minus three is two so that will take us to the point two comma two and then one to the left we need to go down three units so that's the point zero two and then if we go one more to the right we need to go down three units two minus three is negative one so we'll get this point here and another one here and if we go one more to the right and one to left we need to go down three more units so the graph is going to look something like this so now you know how to graph absolute value functions