in this video I will review absolute value functions and I'm going to review them according to this uh plan that I laid out in one of the previous videos called uh pre-calculus review and the plan is to review the prerequisites first and then review all these properties and features um and let's start with the prerequisites first we'll talk about operations and um there aren't too many prerequisites here in terms of arithmetic but we do have the operation of absolute value after which the functions are named so you have to be able to work with absolute values and that means you have to be able to evaluate absolute values and you have to be able to use the properties of absolute values if necessary now in regards to Expressions you have to be able to kind of pull out that negative sign from the absolute value sometimes so you should know that um absolute value of negative 2x it will be the same as the absolute value of 2x there's no need for a negative sign inside of the absolute values now when it comes to equations for example uh when you need to find the x-intercepts of the absolute value function you may end up with the absolute value equation which you absolutely need to know how to solve and how do we solve absolute value equations we use these uh the property and we can call it the absolute value property I guess that is if what you have on the right side is positive then what's inside of the absolute value is either equal to that or the opposite of that so that is if the right side is positive if the right side is negative then you have no Solutions why because absolute value represents distances and distance cannot be negative so absolute value equals to something negative has no Solutions on the other hand if absolute value is equal to zero that can only be due to what's inside of the absolute value is equal to zero which then if you solve it will give you the solution set so this is again a brief overview of how to solve absolute value equations if you need more details please let me know in the comments but again this is what you've studied in algebra now I'm just kind of summarizing it for you now that we have reviewed the prerequisites uh we're moving on to answering all of these questions and let's start with the form when it comes to the form absolute value functions um are the such that can be written like this and if you set H and K equal to 0 and a is equal to 1 you end up with the most basic the simplest absolute value function whose graph is shown right over here now to produce this graph you can throw some points in but at some point you should just remember that absolute value um is the same as the identity function when X is greater than or equal to zero an absolute value is the opposite of that when X is uh negative now it's also important to remember that all other graphs can be produced by using linear Transformations however you will never have to apply a horizontal compression or a horizontal uh flip because of the property of the absolute value uh that allows us to as shown in this example to factor out anything negative and any positive number out of the absolute value so you can simplify this absolute value function uh to look like this and as you can tell there is no need to interpret uh what's inside of the absolute value using linear Transformations you can just do the translations and vertical stretch all right as I said uh we use linear transformations to graph most of the absolute value functions and how do we do that first we plot the vertex uh by reading H and K then we plot the points closest to the vertex using the uh that coefficient in front of the absolute value and the signs so it may open up or down and then we draw the v-shape passing through those uh points now depending on the um combination of values that you get for the leading coefficient or the sign in front of it you may end up with two x-intercepts graph that opens up so it's going to look something like this one x-intercept that looks something like this and no x-intercepts that looks something like this on the other hand you may end up with two x-intercepts with the graph that opens downwards uh one and x-intercept when the graph that barely touches the x-axis and no x-intercepts if the graph opens down below the x-axis now that you have the graph in front of you you should be able to answer questions about the domain and the range the domain of an absolute value function is all real numbers without any constraints and the range of an absolute value depends on again the combination of the linear coefficient and the vertex so if the coefficient is positive then the range is all the values from the y coordinate of the vertex going up and when the coefficient is negative the ranges all the values from negative Infinity up to the y coordinate of the vertex so you should be able to summarize that uh off the top of your head it's very easy to uh produce uh to answer those these questions are very easy to answer um talking about intercepts when we're talking about the y-intercept uh it's pretty much evaluating the function at zero when we're talking about x-intercepts we're going back to solving absolute value equations so as long as this negative K over a value is positive then you will have two uh solutions to these equations and therefore you'll have two x-intercepts if this value is equal to zero this is when you have one uh x-intercept or when this equation has one solution and again when negative K over a is negative this is when you have no x-intercepts and again based on the combination of those numbers you should know which of these six cases uh you are dealing with just by trying to find the x-intercepts now you should know that absolute value functions do not have asymptotes you should know that absolute value functions are continuous on their entire domain from negative Infinity to Infinity however they're increasing decreasing Behavior Uh changes uh halfway through so if you have a V shape that opens up then to the left side of the x coordinate of the vertex the function is decreasing again this is what the V shape looks like it's decreasing then it's increasing if it's upside down v-shaped and on the left side it's increasing on the right side it's decreasing and you should be able to summarize those properties uh really quick now v-shape is clearly not one to one however however uh absolute value function can be Rewritten as a piecewise function with two linear pieces Each of which is then one to one so for each uh branch of the absolute value function you should be able to find the inverse right so for the right side it's going to be this line with the slope one over a and for the left side it's going to be this line with the slope negative 1 over a which is the opposite of that one which kind of make uh sense so once again the absolute value functions are not one to one but each of its branches has an inverse because each of its branches is one to one now when it comes to symmetry uh absolute value functions do have the axis of symmetry that passes through the vertex and whose coordinate whose equation so it's going to be a vertical line and it's a question will be x equals uh what H I actually have it right over here and um this is what I call Axis of Symmetry and when H is equal to zero then this is when we call the function uh even because when H is equal to zero your graph or your vertex will be on the y axis and then it will produce this symmetry which we call even now some special features of absolute value functions um well it's kind of it's the first time I had to say the words piecewise function so that makes it somewhat uh special but um this whole piecewise uh function uh interpretation also gives us a an additional definition to the absolute value so instead of thinking of absolute value as the distance to the origin we actually think of the absolute value uh this way it's X when X is not negative and it's the opposite of X when X is negative so that makes it somewhat uh special and here's an example of writing an absolute value function as a piecewise function um now let's talk about finding equation and when it comes to absolute value equations so technically we have a an absolute value function that looks uh something like this so technically we have three parameters so we probably need three points and there is only one absolute value function that passes through any three points I guess it's not true that's actually interesting question so if the points are like this then definitely there's an absolutely function but what if the points what if this point wasn't there but if this point was here is there a function I guess it doesn't it doesn't mean that there is an absolute oh no yeah of course there is one right so it's just going to be something like this right so I think it's true so the through any given any three points there is only one absolute value function that passes through those three points however while theoretically this is a easy question practically we really not that interested in answering uh this question so you probably will never see a problem like this now when it comes to Applications uh you have to again remind yourself that absolute value represents represents distance and therefore absolute value functions can represent distances as well so here's an example of an absolute value function that represents a distance from some unknown number uh to three so that gives it one application and absolute value functions are great for practicing linear Transformations because they are easy to graph and uh and therefore serve very well as a playground for linear Transformations like I said so this concludes A Brief Review of absolute value functions if you need more details on any of the slides please let me know