in this video we're going to focus on finding the vertex of a parabola of a quadratic equation given in standard form vertex form and factored form so what is standard form it's this form y is equal to ax squared plus bx plus c so to find the vertex you need to use this equation which will give you the x coordinate of the vertex negative b over 2a now once you get your x value you can plug this back in to the original equation to get the y value so that's what we're going to do right now so let's say if we have this quadratic equation y is equal to x squared minus four x plus three now because this is positive x squared we know the parabola is going to open upward and the vertex is basically this point right there so let's find the x coordinate of the vertex so what's a and what's b a is the number in front of x squared if you don't see a number it's a one b is the number in front of x which is negative four and c is three but we don't need that in this example so we know that x is negative b over two a so that's negative and then b is negative four divided by two times a which a is one negative times negative four is four and four divided by two is two so now we have the x coordinate of the vertex now we need to find the y coordinate so all we're going to do is we're going to replace x with y in that equation so y is equal to two squared minus four times two plus three two squared is four four times two is eight now four minus eight is negative four and negative four plus three is negative one so the vertex is two comma negative one so that's how you could find it given an equation in standard form now let's try another example so let's say if we have the equation y is equal to negative 2x squared plus minus five so feel free to pause the video and work on this problem so a is negative two and b is equal to eight so let's use the same formula x is equal to negative b divided by 2a so that's going to be negative times positive 8 over 2 times a where a is negative 2. so we have negative 8 divided by 2 times negative two which is negative four and negative eight divided by negative four is positive two so we have the x coordinate vertex now let's find the y coordinate by plugging in back into the original equation so y is equal to negative two times two squared plus eight times two minus five now two squared is four eight times two is sixteen negative two times four is negative eight and sixteen minus five is eleven negative eight plus eleven or eleven minus eight that's positive three so the vertex is 2 comma 3. now try these two y is equal to 4x squared minus 12x and also y is equal to 2x squared minus 9. so for the example on the left a is 4 and b is negative 12. so using the formula x is equal to negative b divided by 2a that's going to be negative times negative 12 divided by 2 times 4. so that's 12 divided by 8 which we can reduce it so 12 is basically 4 times 3 and 8 is 4 times 2 so you can cancel a 4. so the x coordinate of the vertex is 3 over 2 it's a fraction so now let's find the y coordinate so it's going to be 4 times 3 over 2 squared minus 12 times three over two now three over two squared if you square three three squared is nine two squared is four now twelve times three is 36 and 36 divided by 2 is 18. and 4 times 9 over 4 the 4's will cancel so you can get 9. so this is going to be 9 minus 18 which is equal to negative 9. so that's the vertex for the first example on the right now let's focus on this example so what's a what's b and what's c a is two b is the number that i mean c is the number that doesn't have an x variable next to it so c is negative nine but what's b in this problem if you don't see an x variable then there's nothing b is zero so in this example it's going to be negative zero over two times two zero divided by anything except zero is going to be zero so negative zero divided by four is just zero so now that we have the x coordinate of the vertex we could find the y coordinate so it's going to be two times zero squared minus nine which is zero minus nine and that's negative nine so for this example the vertex is zero negative nine so that's what you can do if you see a quadratic equation in that form now the next form we need to talk about is the vertex form which is y is equal to a times x minus h squared plus k so for this one you really don't have to do any calculation you could just look at it and see what the vertex is going to be the vertex is h k so here you see negative h and here you see positive h so whatever number you see here you're going to have to switch it here is positive k and this is positive k so you don't have to switch that number so i'm going to give you a lot of examples so you can understand what to do let's say if y is equal to 3 times x minus two squared plus four what's the vertex so whatever number you see here we see negative two just reverse it so this is gonna be positive two and don't reverse the number that you see here so it's going to be 2 comma 4 and that's it that's how you could find a vertex if you're given a quadratic equation in vertex form so i'm going to write out a few examples and in the meantime go ahead and write the coordinates of the vertex so the next one we're going to switch down 4 to negative 4 and this is going to stay the same as negative 5. so it's negative 4 negative 5. for the third one we're going to reverse negative 3 to positive 3 and then negative 6 the k value we're not going to change it and here h is negative 3 and k is positive 7. so now you know how to find the vertex of a quadratic equation in vertex form now what do you do if you're given a quadratic equation in factored form how do you find the coordinates of the vertex i'm going to show you two ways the first way you probably haven't seen before but maybe you have finding x-intercepts if you see an x minus two factor the x-intercept is going to be positive two and for this factor x minus 4 the x intercept is 4. now if you were to graph it we would have a point here and another one here and because this is positive x squared if you were to foil it the graph has to look something like this it turns out that the vertex the x coordinate of the vertex is basically the midpoint of the x-intercepts so if the x-intercepts are 2 and 4 the x-coordinate has to be the average of 2 and 4. so it's going to be 3. now intuitively you can see it but let's say if you get two numbers and you're not sure what the midpoint is just add up the two numbers and divide by two so six divided by two is three so we have the x coordinate of the vertex now we need to find the y coordinate so we just got to plug it into this equation this is going to be three minus two times three minus four three minus two is one three minus four is negative one so the vertex is three negative one now let's get the same answer using another method might be longer but at least we can confirm this answer so i'm just going to rewrite the answer that we have the vertex is 3 1 or 3 negative 1 action now what we need to do is we need to foil x times x is x squared and x times negative four that's negative four x and then we have negative two x and negative two times negative four that's positive eight negative 4x minus 2x is negative 6x so now we have the equation in standard form so we can see that a is 1 because there's no number in front of x squared and b is the number in front of x which is negative six so using the formula negative b divided by two a that's going to be negative times negative six over two times one so that's equal to positive six divided by two which is three and we can see we have that answer here so now let's plug it in to the equation so it's going to be three squared minus six times three plus eight three squared which is three times three that's nine six times three is eighteen and nine minus eighteen is negative nine negative nine plus eight is negative one and so that gives us that point so both methods work you just have to pick and choose which method you prefer but let's work on another example so let's say if we have the formula y is equal to negative two times x minus two times x minus five go ahead and try this one so using the first method we have the x-intercepts as two and five so what's the midpoint of two and five it's going to be a decimal 3.5 now what you could do is you can add the two numbers and divide by two so the midpoint as a fraction is seven over two or three point five so that's the x coordinate of the vertex so now we gotta find the y coordinate of the vertex so let's use the original equation it's gonna be negative two and then times seven over two minus two and then times seven over two minus five now let's get common denominators this is two over one if i multiply this by two over two that's equivalent to four divided by two and if i multiply five over one by two over two it's negative ten over two seven minus four is three seven minus ten is negative three so now i can cancel a two so i have negative times three times negative three so that's going to be positive nine divided by this two so the vertex is seven over two comma nine over two if you want decimal values that's three point five comma four point five now let's get that answer using the other technique so let's begin by foiling x minus two and x minus five so we have x times x which is x squared and then x times negative five so that's negative five x and this is negative two x and finally negative two times negative five which is positive two so next let's combine like terms negative five x minus two x is negative seven x now the last thing we need to do is distribute the negative two so first we're gonna have negative two x squared and then it's negative two times negative seven x which is positive 14x and then negative 2 times 10 which is negative 20. so now we have the equation in standard form so let's find a vertex negative b over 2a so we can see that b is positive 14 and a is negative 2. now 14 we can break that into seven times two so notice that we can cancel a two negative seven divided by negative two is the same as positive seven over two so as you can see we have the x coordinate of the vertex 7 over 2 is equal to 3.5 so let me just make some space here now i'm just gonna rewrite this formula on top so now let's plug in this number into this equation so it's going to be negative 2 times 7 over 2 squared plus 14 times 7 over 2 minus 20. 7 squared is 49 and 2 squared is 4 but i'm going to leave it as 2 times 2. now instead of multiplying 14 by 7 it's easier if you divide first fourteen divided by two is seven and then take that seven multiply by this seven that's going to give you forty nine so fourteen times seven over two is forty nine next i'm going to cancel a 2. so i'm left over with negative 49 divided by 2 and 49 minus 20 is positive 29 which i'm going to write as 29 over 1. now we need to get common denominators so i'm going to multiply this fraction by two over two so it's negative forty nine over two plus twenty nine times two twenty times two is forty 9 times 2 is 18 when you add them that's 58 so negative 49 plus 58 or 58 minus 49 that's 9. so this is 9 over 2 which we do have that here 9 over 2 is the same as 4.5 so both ways work you just have to pick which way is easier personally i think the first method is a lot easier than the second but you can choose so now you know how to find the vertex of a parabola given a quadratic equation in standard form vertex form and factored form thanks for watching you