in this video I will do a brief review of linear functions and I'm going to follow this agenda as I described in the previous in one of the previous videos called pre-calculus review that means what does it mean to know linear functions that means you know all the prerequisites and you can provide details in each of these categories if you need more uh description for each of the categories please watch the previous video in which I provided some justification but we're going to start the conversation with discussion of the prerequisites so when it comes to linear functions uh there there are not too many prerequisites and in terms of operations you have to be able to write a number as a fraction and you have to be able to evaluate this simple linear function for all values m x and B and as you can tell it's not that hard and this could be the reason why linear functions are frequently taught early on even to the younger students so it's almost like a competition how young a student can be now before you teach them linear functions um next let's talk about the types of expressions or types of pre-algebra tasks that one have to be able to perform as a prerequisite to this topic and they're pretty straightforward simple tasks such as Distributing and combining like terms now when it comes to equations when you work with linear functions you have to be able to solve linear equations and that need may arise frequently from uh wanting to find the input that produces a certain output such as when you're looking for an x-intercept and when you do that you naturally get what we call a linear equation and to solve a linear equation uh you pretty much isolate the variable I'm not going to give you the lesson here on how to isolate the variable so if you need help please let me know in the comments all right so we're done with prerequisites now we're moving on to um uh actual conversation about the linear functions and first we're going to start with a form um when it when we're dealing with linear functions the standard form that comes to mind immediately it's the slope intercept form uh and we to write in the slope intercept form means to write a function as f of x equals MX plus b where m is the slope and B is the y-intercept now the simplest linear function the you get is when m and b are both equal to 1 and that function is called the identity function and the graph of that function is shown below uh you can always get that graph by plugging in some values and then plotting ordered pairs now if you have to graph just some general function given to you in slope intercept form then you have to uh write the function if it's not given to you in the slope and it's a form you have to write it in such form and once you um know what your slope and intercept are you start with plotting the y-intercept and then you used this little formula um for the slope to plot the other point by simply going from whatever your y-intercept you go up that's called rise and you go to the right that's called run so if rise is negative then you go down if run is negative then you go to the left once you have those two points you can connect those two points and extend them to form the line now depending on the slope the graph may take one of the following three distinct shapes m equals zero gives you a horizontal line when your m is positive you get something that increases from left to right and when your m is negative you get something that decreases from left to right you should know the effect of the slope or rather the effect of the sign of the slope on the shape of the graph um once you have the graph as I previously stated you should be able to discuss domain and range and domain of linear function is always all real numbers so there are no constraints at all so you can plug anything and you will get something however the range of linear function depends um on the slope if the slope is zero remember when the slope is zero all you get is the horizontal line which is the same output so one all you get is the same output your range is just made of a single value however if m is not zero then you get the entire real number line as the range of the function because as you can tell from these images you eventually going to hit any horizontal line uh in the upward and downward Direction so that means the range is all real numbers now let's talk about intercepts uh the y-intercept can be obtained by plugging in zero but we don't need to do that because we all already know that b is called the y-intercept so it's right there in the slope intercept form however to find the x-intercept you always have to plug in 0 for the output that means you have to solve this linear equation this linear equation is pretty straightforward so you can find the y-intercept um as this formula however I don't have this formula memorized I don't think you should it's always easier to just plug in 0 for the output and solve equation as you need it there's no need to worry about it ahead of time now you should know that linear functions always have at least one y-intercept by nature and the number of x-intercepts really depends on the values assigned to m and b so you have no x-intercepts when m is 0 and B is not zero you have one x-intercept when m is not equal to zero those are kind of the cases you see over here and you have influential many X intercepts when your graph um seats on the x-axis so you should know that um now let's talk about asymptotes asymptotes um uh by Nature well I guess by Nature linear functions don't have asymptotes but also by definition asymptotes are frequently expressed as lines and if it's a horizontal line or slant uh asymptote then it can be viewed as a function so again we may not be dealing with asymptotes when working with linear functions but we may be working with linear functions when working with asymptotes for some other functions um which is important for you to remember now it's important to also remember off the top of your head that linear functions are continuous and when it comes to monotonicity as a consequence of one of the previous facts the following is true when m is negative linear functions are always decreasing we've seen that when m is zero linear functions are always constants and when m is positive linear functions are always increasing it's a very simple direct relation between the slope and the monotonicity now when it comes to symmetry uh there are in general linear functions don't have symmetry but when m is equal to zero functions are even and when B is equal to zero functions are odd so when m is equal to zero you have a horizontal line so there's a natural symmetry over with respect to the y-axis so it's even and when B is equal to zero you naturally have a function that passes through the origin one way or another it's it possess the symmetry that we refer to as if the function is off so these are the two cases and you should be able to remember that now what are the special features of linear functions the the first special the only special feature that comes to mind is that linear functions um are the functions that have constant rate of change so as we move on to calculus rates of change is something that we will be considering and linear function is the function that has the constant rate of change and you can compute that constant rate of change using any two points uh as we have seen previously by using this formula rise over run so this also the formula that can help us with finding the equation so as I said before generally speaking a function with two parameters requires two points um so when two points are given to you you should be able to find the equation of uh this linear function so one way is to find the slope and then use the point-slope formula again I'm not going to give you too many details so if you need me please let me know in the comments uh but briefly speaking once you have the slope and the point you can use the point-slope formula to find the equation of the line however alternatively what we can also do is use those two points plug in the inputs into the function and set them equal to the outputs and by doing that we'll end up with a system of equations in which the variables are m and b it's the it's a simple system of two linear equations which we should be able to solve for m and b and here I give you the general solution but we don't need to remember the general solution you can always just solve the system when you see one and sometimes we have seen those in algebra those types of problems when information about the slope is given to you uh somehow by providing a a a fact uh some fact about the line and its relation to some other line so you've seen that before again if you need more examples let me know now finally let's talk about applications uh there are infinitely many applications for linear functions uh what you need to know is it's the meaning because frequently by the time students reach high level math classes they forget what m and b stand for m stands for the rate of change of the output per one unit of change in the input you just have to remember that and y-intercept stands for the initial value in other words the output when the input is equal to zero so remember that now frequently in applications there are units the units of the output are always I'm sorry the units of the Y intercepts are always the same as the units of the output and the units of the slope are always the units of the output divided by the units of the input so if you remember that then you pretty much all said so this was a brief review of linear functions uh if you need more details please let me know in the comments