in this video we're going to focus on parabolas so let's talk about some equations that you need to know so hopefully you have a sheet of paper with you and a pen to write down some notes so for the parabola on the left this corresponds to the equation y squared is equal to 4px and you would use this if the vertex of the parabola is the origin for the equation on the right its x squared is equal to 4py with the vertex being at the origin as well the focus is to the right where the parabola opens towards so for this parabola it's somewhere in this region p is the distance between the vertex and the focus now if you go p units in the other direction you're going to get something called the directrix the directrix is p units away from the vertex for this particular graph it's going to be x is equal to negative p now when p is positive the graph is going to open towards the right when p is negative it's going to open towards the left for this graph the focus would be somewhere in this region it's going to be p units away from the vertex and then p units below that we have the directrix in this case we have a horizontal matrix thus the equation will be y is equal to negative p as opposed to x is equal to negative p now going from the focus to the curve that's the distance of 2p and going in the other direction is a distance of 2p so the segment that connects these two points on the curve and that passes through the focal point that segment is known as the lattice rectum the lattice rectum has a length of 4p it's basically the focal diameter of the parabola now when p is positive the parabola is going to open upward if you have this equation and when p is negative it's going to open downward so that's a simple introduction into parabolas but let's begin working on some problems by the way for those of you who want access to the full version of this video feel free to take a look at the links in the description section below this video so let's go ahead and continue graph the equation shown below so this equation is in the form x squared is equal to 4 p y with the vertex centered at the origin there's no h and k in this equation which we'll talk later about that in the video now the first thing we need to do is calculate p 4p is the number in front of y so we're going to set 4p equal to 8. dividing both sides by 4 p is going to be 8 divided by 4 which is 2. now once we have our p value we can go ahead and graph the equation so first let's plot p the vertex is at the origin and going 2 units up we get the focus which is here so the focus is at zero comma two now once you have the focus what you wanna do is you wanna go two p units to the right and 2p units to left if p is 2 2p is 4. so we're going to travel 4 units to the right and 4 units to the left to get 2 additional points and then we can draw a rough sketch of the hyperbola so keep in mind this here is 2p and this part is 2p now let's write the equation of the directrix so we're going to go p units below the vertex and then we're going to draw a horizontal dashed line so that line is that y is equal to negative 2 and that's the equation of the matrix so that's it for this problem that's how we can graph this particular parabola number two graph the parabolic equation shown below find the coordinates of the focus and write the equation of the directrix and determine the length of the lattice rectum so our general formula is going to be y squared is equal to 4px so let's begin by setting 4p equal to the number in front of x which is 2. so four p is equal to two dividing both sides by four we get p is two over four which reduces to one over two so now that we have the value of p and we know that the vertex is centered at the origin let's go ahead and graph it so for this equation the graph is going to open towards the right since p is positive the vertex is at the origin and let's put some points here since p is small i'm going to space out each point now p is one half so point five units to the right of the vertex will be the focus now i'm going to travel up 2p units and down 2p units if p is a half 2p is going to be 1. so the next point will be here and the other point will be here so to graph it it's going to look something like this let's see if i can draw a much smoother graph we'll go with that so the coordinates of the focus is the x value is one half and the y value is zero now going one half units in the other direction we're going to get the equation for the directrix and that's at x equals negative p or x equals negative one half so now we have the equation of the directrix and we have the coordinates of the focus the last thing we need to do is determine the length of the latest rectum or the latus rectum and it's basically the focal diameter connecting these two points in the curve so we know this part is 2p this part is 2p thus the left of the lattice rectum is always going to be four p so p is a half four times a half is two so the answer to the last part of the problem is two units long number three match each equation to one of the graphs shown below so let's start with this one x squared is equal to four y when you have y and x squared that's a graph that's going to open up or down now there are no negative signs here so this one is going to open up so therefore this corresponds to answer choice a the next one y squared is equal to 4y when you have y squared and x to the first power this is going to open either to the right or to the left but since there's no negative sign it's going to open to the right so this is b now for c y squared equals negative 4y it's opposite to this one so if this one opens to the right the y squared equals negative 4y is going to open to the left the last one d x squared equals negative 4y is opposite to a so this one is going to open downward so make sure you understand how to associate the correct orientation of the parabola with the appropriate equation because you need to know this if you're going to study four tests and this is going to be very important whenever you need to graph a parabola especially the harder versions so just to review when x is squared and y is not it's going to open either up or down so for this one when it opens up we have positive 4y when it opens down there's going to be a negative sign so this is x squared is equal to negative 4y now when y is squared and x is not it's going to open either to the right or to the left p is positive it's going to open to the right when p is negative it's going to open to the left so make sure you understand this number four write the standard form of the equation for the parabola with the given conditions so we're given the coordinates of the focus and the equation of the directrix so let's go ahead and plot the information that we have here so let's start with the focus so the focus is at negative three zero so there it is and then we have the directrix at x equals positive 3. what we need to find is the vertex the vertex of the parabola is going to be the midpoint between the focus and the directrix the focus is at positive three the directrix is at negative three so when you average those two numbers you'll see that the vertex is going to be at the origin zero zero so to draw a rough sketch of the graph we know that it's going to open toward the focus away from the metrics so this graph is going to open towards the left now in order to write the equation we need to calculate p p is the distance between the vertex and the focus and also between the focus and the directrix in both cases we can see that p is three units long now is p positive 3 or negative 3 the fact that the hyper i mean not the hyperbola but the parabola opens towards the left towards the focus means that p is negative so p is negative three in this case this part here is going to be 2p and that part is going to be 2p below the focus 2p is 6. so if you want to draw an accurate sketch this point here should be at a y value of six and this point here should be at a a y value of negative six now my sketch is not drawn to scale but those would be the points that is part of the lactus rectum so now that we have our p-value we can now write the equation so what's the general form of the equation for a parabola where the vertex is at the origin but it opens to the left this is going to be y squared is equal to 4px for horizontal parabola that opens to the left or to the right now all we need to do is plug in p so p is negative three four times negative three is negative twelve so this here is the final answer this is the standard form of the equation for the parabola with a focus of negative 3 comma 0 and a directrix of x equals positive 3. so the answer is y squared is equal to negative 12 x number five identify the coordinates of the vertex and focus write the equation of the directrix graph the parabola calculate the length of the latus rectum and determine the domain and range of the function so let's begin so this equation is in the form y squared is equal to 4px now we know that equation has its vertex at the origin and it opens to the right if p is positive and we have a directrix here p units away from the vertex now we do have some numbers associated with x and y so this particular parabola has been shifted away from the origin the vertex was zero comma zero now it has been shifted to some point h comma k when it's been shifted this equation changes to this it becomes y minus k squared is equal to 4p x minus h where h and k are the coordinates of the vertex now the focus which was here initially it initially had the point p comma zero but now once you add the new vertex to it once you add h comma k the new focus becomes h plus p comma k as for the equation of the directrix it was x equals negative p but once you add h to it it becomes x is equal to h minus p so let's go ahead and work on this problem hopefully you wrote those down but as long as you understand it you can get everything that you need but let's write this equation y minus k squared is equal to 4p x minus h and let's find the vertex so the vertex is going to be h comma k so looking at the number associated with x it's a plus two to find h simply reverse the sign so this is going to be negative two and the number in front of y is negative three so reverse it to positive three and now we have h and k so h is negative two k is three now the next thing we need to do is find the focus and the vertex but let's start with a graph so the vertex is at negative 2 comma 3 which is here and now let's calculate p so 4 is equal to 4p dividing both sides by four we get p is equal to one so now that we know what p is and p is one so we know that this is going to open towards the right if we travel one unit to the right we'll get the focus and then we're going to go up 2p and down 2p so if p is 1 2p is 2 so that will take us to this point and that point and now we can graph let me use a different color so this is how our parabola is going to look like now let's travel p units to the left p is one so the directrix will be one unit away from the vertex so we can clearly see that the equation of the dimetrics will be x is equal to negative 3. any time the directrix is vertical it's going to be x is equal to a number if it's horizontal it's y is equal to a number the formula for calculating the directrix is h minus p for this type of shape but if you don't want to memorize the formula you could just look at the graph and see what the answer is h is negative two p is positive one so negative two minus 1 you get negative 3. for those of you who want to see how to use that equation now the focus to find the coordinates of the focus we could just look at the graph we can see it has an x value of negative 1 and a y value of three using the formula it's h plus p comma k for this type of horizontal parabola h is negative two p is one k is three so negative two plus one is negative one so we get that the focus is negative one comma three so right now we have the coordinates of the focus we have the equation of the directrix we also have the coordinates of the vertex and we have the graph of the parabola the next thing we need to do is find the length of the lattice rectum so that's going to be the distance between those two points the length of the latus rectum as we've considered before it's always going to be equal to 4p and since p is 1 this is going to equal 4. the last thing we need to do is determine the domain and range of the function so for the domain we're looking at the x values from left to right the lowest x value is the x coordinate of the vertex which is negative two since negative two is part of the graph we include it with a bracket now this is going to go all the way to the right all the way to positive infinity so the domain is from negative 2 to infinity now for the range we're going to focus on the y values from the bottom to the top this graph will keep on going forever it's going to go to the right and it's going to go down so it's going to go down all the way to negative infinity and you could follow all of the y values along this curve as it goes up to positive infinity so for a horizontal parabola that opens to the left or to the right the range is always going to be real numbers if you have a vertical parabola that opens up and down then the domain will be all real numbers negative infinity to positive infinity number six identify the coordinates of the vertex and focus and basically do everything that we did in the previous problem so the standard form of the parabola for this problem is going to be x minus h squared is equal to 4p times y minus k and the vertex is going to be at h k so let's begin by finding the vertex first here we see negative 3 so we're going to have positive 3 for x for y we see negative 2 so we're going to change it to positive 2. so that's the coordinates of the vertex for this particular parabola so h is three k is two now let's calculate p negative eight is equal to four p let's divide both sides by four and so p is going to be negative eight over four which is two now let's go ahead and sketch a graph for this parabola so let's begin by plotting the vertex which is at 3 comma 2. now what direction will the graph open will it open to the right will it open to the left will it open up or will it open down what would you say the first thing we need to pay attention to is which variables are squared and which ones are not x is squared y is not so whenever you have y equals x squared this is a parabola that opens up or down and since p is negative this is a parabola that's going to open in a downward direction so p is negative two we're going to go down two units and this will give us our focus which is here so we can see that the coordinates of the focus is at three zero the formula to calculate the focus for this type of parabola that opens up or down it's going to be h comma k plus p h in this example is 3 k is two and p is negative two two plus negative two is zero so we get the point three comma zero so now that we have the coordinates of the focus let's find the other points that we need in order to graph this parabola so we're going to travel two p units to the right and two p units to the left p is negative two so two p is going to be negative four but we're gonna use the absolute value of that so the absolute value of two p is four so we're going to travel four units to the right from the focus and then four units to the left and now we can sketch our parabola so now let's go up p units this will give us the equation of the directrix so here the y value is two here it's going to be four thus the equation of the directrix will be y is equal to positive four when dealing with a vertical parabola the equation that you need to calculate the directrix is this y is equal to k minus p k is two p is negative two so you get two minus negative two which is the same as two plus two and so you get y is equal to four now let's determine the length of the lattice rectum so the lattice rectum is the focal diameter that connects two points on the curve and passes through the focus so we know this part is 2p the other part is 2p so that gives us the total length of 4p now p is negative and the left of that segment we're just going to assign a positive value to it so technically we need to say that the length of the lattice rectum is the absolute value of 4p so that's the absolute value of 4 times negative 2 which will give us 8. so the length of the lattice rectum in this example is 8 units long now let's focus on the domain and range starting with the domain for a vertical parabola it's going to be all real numbers if you analyze it from left to right the lowest x value is negative infinity the highest x value is infinity and x could be any number in between those two extremes so for any vertical parabola the domain is always all real numbers the range is different though so for the range we'll analyze it from the bottom to the top the lowest y value this can go down forever so the lowest y value is negative infinity a vertical parabola like this one has a maximum if it opens downward it has a minimum if it opens upward the maximum is basically the vertex the y coordinate of the vertex is two so the range is going to be from negative infinity up to the y coordinate of the vertex which is two so that's how you can determine the domain and range for a parabola number seven write the equation of the parabola in standard form identify the coordinates of the vertex and focus write the equation of the directrix and graph the parabola so we have the equation in non-standard form so to speak we need to put it in standard form how can we do this well we're going to have to use a technique called completing the square but first we need to know what type of parabola we're dealing with notice which variable is squared and which one is not x is squared y is not so therefore we need to put this equation in this format x minus h squared is equal to 4p times y minus k so how can we turn this equation into something that looks like that well we could see that the x variables are on the left y is on the right so that gives us some guidance in terms of what we need to do here sometimes this software acts up what we're going to do is we're going to keep the x variables on the left side so we can mirror what we have there everything else we're going to move it to the right side so on the left side of the equation we're going to have x squared minus 4x leave a space negative 8y when we move it to the right side it's going to become positive 8y and negative 4 will change to positive 4. now we're going to complete the square so looking at this number we're going to take half of it and then we're going to square it so i'm going to add 2 squared to the left side 2 squared has a value of 4. since i've added 4 to the left side i need to do the same to the right side such that the equation remains balanced now we can go ahead and complete the square so it's going to be x minus 2 squared on the right side we can add 4 plus 4 which will give us eight so we have eight y plus eight next we could factor out an eight from eight y plus eight so this becomes y plus one so now we have the equation of the parabola in standard form so i'm just going to erase what we have here and then i'm just going to rewrite our new equation here so now that we have the standard form of the parabola we can go ahead and find the rest of the answers we're looking for so let's start with the coordinates of the vertex here we have negative 2 in front of x we're going to change this to positive 2. here we have plus 1 we're going to make this a negative 1. so that's the coordinates of the vertex h is 2 k is negative 1. now let's calculate p we're going to set 8 equal to 4p so if 8 is equal to 4p 8 divided by 4 is 2. so we get that p is equal to 2. now let's go ahead and sketch a graph most of the graph will be on the right side so let's begin by plotting the vertex so it's at 2 negative 1 which is here and then p is 2. so what we need to do is determine what direction the parabola is going to open is it going to open up down right or left what would you say well looking at this x is squared y is not so when you have a general graph y equals x squared that's a problem that opens up if p is positive which it is so we know the graph is going to go in this general direction so now we know where the focus is going to be p is 2 so going 2 units above the vertex takes us to this point so that's where the focus is so we could say that the focus has an x value of 2 and a y value of 1. now let's confirm that with this equation the equation for the focus when dealing with a vertical parabola that's one that opens up or down it's going to be h comma k plus p so we know h is 2 k is negative one plus p p is two negative one plus two is one so we've confirmed that this answer is correct so that's the coordinates of the focus now let's find some other points so to get the other points we need to travel 2 p units to the right two p units to the left so if p is two two p is going to be four so going four units to the right of the focus takes us to this point and four units to the left takes us to that point so now we can sketch a rough graph of the hyperbola it doesn't have to be perfect but at least this will be a good approximation so now let's find the equation of the directrix so let's go p units in the other direction and so the direction i mean the directrix rather will be here so this is at a y value of negative one and here this is at a y value of negative three so we can say that the equation of the directrix is y is equal to negative three now let's confirm it with the equation for directrix when dealing with a vertical parabola it's y is equal to k minus p k is negative 1 minus p which is positive 2. negative 1 minus 2 is negative 3. so you want to be familiar with these equations it's good to know how to use them so that's why i want to go over confirming our answer with those equations i want you to be familiar with those equations for the different types of parabolas that you may encounter so that's basically it for this problem we did everything that we needed to do you