[Music] [Applause] [Music] okay so we are going to continue our discussion for lesson 1.2 on topic nothing it's all about the equation of a parabola but this time we're going to have vertex at h k so we begin our discussion with the upward parabola so the parabola vertex of h k and is greater than 0 is given by the standard equation the quantity of x minus h squared is equal to 4a times the quantity of y minus k where the vertex is at the point h k the axis of symmetry is at x is equal to h the focus is at h and k plus a the directrix is equal to y is equal to a minus a and lastly for the lattice rectum okay for the coordinates of the lacrosse rectum it will be given by the coordinates h plus 2 a and k plus a while the other coordinate is given by h minus 2 a and k plus a so this will be the summary of the integral parts when we have the parabola that is opening upward with vertex at h k so for the illustration and it is the left side so notice this is the parabola the blue the blue curve so on vertex nothing is at h k the focus is at h comma k plus a the axis of symmetry this one is x is equal to h the directrix is this yellow line the horizontal line y is equals of a minus a and this will be the x and y axis the white lines okay let's have the first example example number one the quantity of x plus five squared is equal to two times the quantity of y plus two so the direction is already given we need to sketch the graph and indicate the focus directrix vertex axis of symmetry and the coordinates of the endpoints of the lattice rectum so first we begin we analyze the given since i'm given a squared x term so it picks at the end this will be either up or down and since 4a or since the value of a is positive that is the numerical coefficient of 2 so the parabola will open upward then what will be the vertex the vertex is at the point h comma k so this will be h and this will be your value of k so h k will have a value of negative five negative two so this one so nandita in vertex net next how about the axis of symmetry the axis of symmetry dao is located it is a line x is equal to h so whatever is from x coordinate on vertex union body and axis of symmetry morita so in short the axis of symmetry is given by the equation x is equal to negative five next for the focus for the focus we have h then comma k plus a so what's the value of k that is already given as negative two then for ac equals 2 then we divide both sides by 4 to get the value of a so on a is equal to one half so now which is negative two so now we have negative five comma negative two plus one half so adding k plus a so what is negative two added to one half or negative two plus one half so this will give us negative five and negative three over so 92 in focus not n okay next for the directrix it will be given by the equation y is equal to k minus a so by the meeting so opposite side so y is equal to k minus a so if it's a b and that will give us y is equal to negative 2 and k minus imbalance in one half then anatomy evaluate so what is now y is equal to negative 2 minus 1 up so this will give us the value of y that is equal to negative five over two okay for the last one we need to get the coordinates of the lattice rectangle so the coordinate of the lattice rectum is given by h plus 2a and h minus 2a same y coordinates opens upward that's the same line beyond endpoints nonetheless rectum and the nd focus so i am following this uh given for the lattice rectum by substitution so we have we need to get the value up to a purse so 2a so it picks a b and that is 2 times 1 half so what is 2 times 1 up that will give us 1. so i think that of the data selecto carbon side and focus okay so for this h plus 2 a so that will be given by negative 5 because atom h i think 2 a naught and has a value of one so plus one monotheito then we copy the y coordinate in focus then cytoman ends up in that spectrum that will be given by h minus two a it picks a b a negative 5 minus 1 couple same y coordinate and focus so if i can evaluate nothing to so this will give us negative 5 plus 1 so that is negative 4 while on the other side we have negative 5 minus one that is negative six so for the coordinates of the lattice rectum so this will give us negative four and negative three over two and negative six and negative three over two so once that integral parts and then we can draw the parabola okay so we begin first by locating the vertex so the vertex is at the point negative five negative two so liters of partition plane nothing's the right side so this will be the vertex location next we have the axis of symmetry the axis of symmetry is a line that is x is equal to negative five okay next we plot the focus the purpose is at the point negative pi and negative three over two negative three over two is equivalent to negative 1.5 so this will be the focus and next after that we have the directrix the directrix is at twice equals to negative 5 over 2. so negative negative 1.5 mm focus on directx minus a negative 2.5 so nandito see directrix then the endpoints of the lattice rectum we have negative 4 and negative 3 over 2 so a negative six and negative three over two param from the focus plus one minus one nangian and yarijan so we have these coordinates since all the integral parts are plottedness adding cartesian plane then we can draw a smooth curve connecting the vertex and the endpoints of the lattice rectum parabolium graph parabola so this will be the graph of the parabola in example number one next example hey how about a downward parabola so it means we have a negative coefficient a so what is the standard equation again that is given by the equation the quantity of x minus h squared is equal to negative 4a times the quantity of y minus k the vertex still at h k with adam and evago your axis of symmetry parenthesis and operands align the x is equal to h the focus is now given by the coordinate h and k plus a number so the directrix is given by y is equal to a minus a so that's still the same and the endpoints of the lattice rectum detector consider that not in your absolute value no adding to a okay that is rectum so we need to get the absolute value then add and subtract [Music] so let's have an example given the equation the quantity of x minus v squared is equal to negative 8 times the quantity of y plus four so again the direction is sketch the graph indicate the focus directrix vertex axis of symmetry and the coordinates of the endpoints of the lattice rectangle so first we inspect the event squared view x term so it picks up the parabola it's either opening upward or downward now looking at the numerical coefficient detail this is negative eight so since this is negative we all know that the opening is downward next for the vertex the vertex is at h k so at imbilinum h at the environment so that becomes three and negative four so nothing in vertex not and at the point three negative four the axis of symmetry is given by x is equal to h so whatever your x-coordinate or h-bar unit ending axis of symmetry so x is equals to three the opposite is given by the equation by the coordinates h and k plus a so big sub h and copy audit [Music] [Music] so that is plus negative two so what is now negative 4 plus negative 2 so that will give us a focus of located 3 and negative 6. next for the directrix we have y equals a minus a so we have y is equal to negative four adding volume of a then minus a minus negative two so negative times negative is positive so i'm getting negative 4 plus 2 so you direct x naught and i'm appearing y is equals negative 2. okay so last integral partner is a subnet is the lattice that come or the coordinate of the lattice that too complete into a marina time by the way to know a so easily we can get the value of 2a so your absolute value to a that is 2 times negative 2 is negative 4 absolute value of negative 4 is 4. so it picks up the end we need to add and subtract 4 then it's adding focus paradigms h value okay then plus and minus four so this becomes seven this becomes negative one so in coordinate and that is rectum is at the point seven negative six and negative one negative six so again completed integral parts straight vertex that is 3 negative 4. next we have the axis of symmetry the axis of symmetry is x is equal to three next for the focus we have three negative six and it imposes net and sub about the vertex because the opening no parabolic and is downward so that is three negative six next on directrix vertex so that is y equals to negative two and [Music] so we need to plot negative one negative six and seven negative six so this will be the coordinates of the lattice rectum now we connect the vertex to the lattice rectum parameter so this will be the query okay so this will give us the graph for the example number two that is the quantity of x minus 3 squared is equal to negative 8 times the quantity of y plus 4. so now we're done with the parabola that opens upward and downward so next is for me to give you example of those parabolas who open sideways so in parabola that opens to the left and the parabola that opens to the right if this is positive the parabola opens to the right at y term in a squared so sideways in parabola then okay so the standard equation for the parabola that opens to the right is given by the equation the quantity of y minus k squared is equal support a times the quantity of x minus h in vertex no parabola is still at h k the axis of symmetry is now at the line y is equals okay so it is then the focus again coconut netting okay coordinates none in and it is a vertex then little entire x-coordinates the entire each body maga adnan a so that is h plus a and k so the directrix again complex agnosta cocos minus under partners directrix that is given by h minus a then for the lattice rectum coordinates [Music] so then above in focus vertex the directrix the axis of symmetry so completely okay let's have example number three given the equation the quantity of y plus six squared is equals four times the quantity of x minus three so now observe one another in a squared y term so it picks up the end it's either left or right parabola and since inversion on numerical coefficient nothing little i positive so if it's a b and the parabola opens to the right so opening here is right side hey the vertex is at each k so at the young age you know d is x term so on h naught and a three while again at an i negative six so in vertex i now have point three negative six so in axis of symmetry nathan at all y is equals okay so component in negative sixty two vertex segment [Music] okay next focus is at the point h plus a comma k so is solving a by using jung at a numerical coefficient beta so for ac equals a power we divide both sides by four so we'll get a is equal to one so once we get the value of a so predict add coulomb h so we have h plus a and k so it will be given by three plus one and six and we'll have four negative six the directrix is h minus a so diameter of a so x is equal to three at the new h minus a minus one so three minus one will give us x is equal to two then for the event of the lattice rectum or particle coordinates of the lattice rectum and nothing to a so dynamic value in the thyroid and a at times so we'll get 2 a is equals 2 times 1 and that will give us positive and negative 2. so subtract and do so using these points by substitution so we have four and negative six plus two the apples so negative six plus two is negative four negative six minus two is negative eight so your coordinates now endpoints and that is rectum is now given by four negative four and four negative eight so once then we let the heart and integral parts draw the parabola begin with the vertex then a platinum eliminating y c equals negative six so this would be the axis of symmetry then it platinate the new focus at four negative six so make it an attenuating parabola nasa write number text a then the directrix is at x is equal to two so this will be a vertical line then the coordinates of the the endpoints of the lattice rectum is 4 negative 4 and 4 negative 8 so once we have the vertex and the coordinates of the lattice rectum or the endpoints of the lattice rectum so we can draw the parabola so that will be all for example number three so a right parabola or a parabola that opens to the right okay last one we have the parabola that opens to the left so the parabola vertex at h k n a is less than zero so that will be given by the equation the quantity of y minus k squared is equal to negative four a times the quantity of x minus h the vertex is at h k the axis of symmetry is at y is equal to k so same paranormal the focus is at h plus a and k the directrix is at x is equal to h minus a solana mandinhar is nabago right parabola while the lattice rectum we need to consider the absolute value to a so in the main difference n okay let's try this problem four the quantity of y minus two squared is equal to negative six times the quantity of x minus one so we need to sketch the parabola and indicate the following focus the electrics vertex axis of symmetry and coordinates of the endpoints of the lattice electron first we begin by inspection [Music] and since the numerical coefficient it is negative so it picks a b and the parabola opens to the left next for the vertex the vertex is located at h k so this is h and this is k so the value is at one two the axis of symmetry is at the point is at the line y is equal to k so we get the x coordinate the y coordinate or decay value in vertex data so that is y c equals a two next we need to solve for the value of a parameter h so considering a numerical coefficient d to the negative 6 so we have 4 ac equals negative six we divide both sides by four so a has a value of negative three over two so now we can have the coordinate of the focus so the focus is at the point 1 plus negative 3 over 2 n 2 so that is h plus a h plus a so what is now 1 plus negative 3 over 2 or 1 minus 3 over 2 so 1 minus 3 over 2 will give us negative 1 half n 2 for the directrix we have h minus a so we have x is equals to 1 minus negative 3 over two so this becomes negative times negative positive so omega add line n one plus three halves so that will give us five over two next for the coordinates of the lattice electron phonemon and antenna absolute value into a so now we have the absolute value of two a that is two times negative three over two or positive negative three along so big sub b and we need to add and subtract 3 both sides known for us not n or it will be using these coordinates so by substitution we have negative one half and two plus three tap was negative one half and two minus three so two plus three is five two minus three is negative one so final answer nothing's end points is negative one half and five and negative 1 up and negative 1. okay now let's make the graph of the quantity of y minus 2 squared is equal to negative 6 times the quantity of x minus 1. so first we begin with the vertex at one two then we have the axis of symmetry at y c equals two then the focus at the point negative one half and two so the directrix is at the line x is equal to five over two and the endpoints of the lattice rectum so we have negative one up and five and negative one half and negative one so now connecting the vertex and the lattice rectum then extending it to the left side so we have here the parabola okay so that will be all for this topic for your assignments it will be given as assignment number four so you need to draw or sketch the graph and also indicate the focus directrix vertex axis of symmetry and the coordinates of the endpoints of the lattice right tom of these five items so do it in your graphing notebook so that will be all thank you and see you next meeting you