hello everyone welcome to senior pablo tv so after discussing our circle let's now proceed in our second kind of conic section the parabola before watching this video please make sure that you subscribe in our channel and click our notification bell in order for you to be updated in our upcoming videos the parabola first let us define the parabola is the locus of all points whose distance from a fixed point our f and a fixed line for the l not passing through through f are the same the fixed point is called the focus while the fixed line is called the directrix so when you were in grade 9 you already discussed the parabola but in your grade 11 there are [Music] mathematical terms that you will encounter just like our focus and directrix later we will discuss that in our graph to better understand we know that parabola is a curve can be opened upward sideward the left opens to the right or opens downward those are the graph of parabola note the line that passes through the focus and is perpendicular to the directrix is called the axis of symmetry of the parabola and the lattice rectum is the line that passes through a focus perpendicular to the axis of symmetry okay to better understand those mathematical terms let us locate it in our cartesian plane and our graph let's say this is our parabola okay this curve and this curve opens upward we have here the focus that is the fixed point so where is our focus here so this is our focus this is our focus okay where is the directrix while the fixed line is called the directrix so this is our directories a line fixed line not passing through the focus are the same this is our directrix next term is the axis of symmetry the line that passes through the focus and is perpendicular to the directrix a line passes through the focus this is our directrix that is perpendicular to our directrix so this is our axis of symmetry axis of symmetry and the lattice rectum is a line that passes through the focus perpendicular to the axis of symmetry so this line passes through the focus and it's perpendicular so this is the lattus rectum lattus in parabola we're going to find the length of lattice rectum okay so that is the definition and different terms in our parabola now we're going to find the axis of symmetry directories focus and lattice rectum of the parabola using the given equation station and now let's have this example determine the opening vertex focus directrix axis of symmetry and endpoints of lattice rectum of the parabola with the given points then graph so we're going to find the opening opening opening whether it is upward downwards to the open to the left or open to the right then the vertex focus directrix axis of symmetry and endpoints of the lattice rectum given the parabola x squared is equal to 12 y okay let's have first the standard equation of parabola so that we can easily solve this problem so if our equation is at vertex point okay this is the vertex form of our of the equation of the parabola vertex form x squared is equal to 4cy and if this is the given x squared is equal to 4cy that means it opens upward because c is greater than zero and if it is in h k form so this will be the equation of the parabola and x squared is equal to negative four c y at vertex form take note c is less than zero that means open downward and this will be the equation for hk and opens to the right to the right if c is greater than zero then open to the left if c is less than zero so we have different formulas here that we are going to use for c is the length of the lattice rectum for c this one and c is the distance from the vertex to the focus from the vertex to the directrix so later on after graphing our problem you're going to understand this then 2c the distance from the vocals to the endpoints of the lattice rectum using this table we're going to find first the opening so x squared is equal to 12 y so that is in the vertex so our vertex is zero zero so vertex we can look at that zero zero then we are in this form c is greater than zero or the p is greater than zero that means our opening is up more next find the focus directories and axis of symmetry so first let us find the focus to find the focus let us find first the value of c so to find the value of c we have here 4 c is equal to this is our 4c which is 12. for c oh we need to find c so we need to divide by 4 so c is equal to three twelve divided by four that is three that means from our vertex that is three units up to our vertex so our focus is zero positive three this is our focus now let's plot in our partition plane we know that our vertex is zero zero so let us locate 0 0 in our origin with our focus we know that it's open upward so that means it is located up of our origin so zero three okay the focus one two three this will be our friends okay this is our focus focus is 0 and this is the vertex now let us find directrix so we have directories directrix is perpendicular to our axis of symmetry so that is three units below our vertex three units below so that is zero going down one two three negative three but directrix is a line so we're not going to plot the points we're going to find the equation so y is equal to negative 3 because directrix is a line okay y is equal to negative 3 so if we're going to find y is equal to negative 3 it's here negative 3 is located here so just create a line passing into negative 3 this will be our directrix next after getting the directrix we're going to find the axis of symmetry axis of symmetry axis of s let's say axis of x axis of x s or symmetry is perpendicular to the directrix so it's here passes through the focus so that is x our value of x that is zero so it's here along the x axis now we're going to find the endpoints of the lattice rectum of the parabola so since we have c here and we have the formula 2c the distance from the focus to the endpoints of the lattice rectum so 2c substitute so 2 times our c is three so from the focus to the end of a lattice rectum is six units so we're going count six units to the left the left and to the right so one two three four five six and one two three four five six so our points are six and one two three three here we have negative six and positive three now connect our points from the endpoint of the lattice rectum going to our vertex then going to the another endpoint of the lattice okay this is now our parabola okay that's on how to find the opening vertex focus directrix and the length of the lattice rectum or the endpoints of the lattice rectum now if your teacher asks you what is the length of the lapis rectum so the length of the lattice rectum is 4 times c that is 12. so length of lattice rectum is to all the units that's it that's the parabola now your turn i want you to answer this equation our assignment i'm going to write there is this [Music] the equation is y squared is equal to negative 8 x your assignment y squared is equal to negative 8 x so what if the given equation is in the or our vertex is at h k so how we're going to use or to find the opening vertex focus directrix axis of symmetry and endpoints of the lattice rectangle so again we're going to use this table to easily easily solve the equation so our problem is determine the opening vertex focus directrix axis of symmetry and endpoints of the lattice rectum of the parabola with the given equation that graph so our equation is quantity x plus 1 raised to 2 is equal to eight quantity y minus three first we're going to determine the opening opening whether it is upward downward open to the right or open to the left our basis is our c so in this case c c is positive that is greater than zero that means it's in this form that is upward so opening up or next we're going to find the vertex the vertex is our h k so from the given just equate to zero the x x plus one is equal to zero and for our y y minus three is equal to zero this will be our h and our k so x is equal to negative one y is equal to positive three therefore our focus is negative one for our x coordinate and for our y coordinate is positive three negative one positive three okay now let us locate negative one positive three so negative one plus the one two three it's here negative one positive three next we're going to find the focus so focus our focus or the p that is equal to our 4c so we're going to find that 4c is equal to here 4c is 8. so divide by four four c is equal to positive three that means from our vertex we're going to locate two units up because the opening is upward so 1 2 this will be our focus so our focus is negative 1 then our y is one two three four positive four this is our focus let us box our final answers opening okay opening vertex and focus next directrix directrix since we have c so we're going to locate two units down our vertex this is our vertex and this is our focus two units down so one two okay so that is directrix that is the value of y our equation is y is equal to our y here is positive one because our directrix is a line okay y is one box for final answer this is our directories directories and next axis of symmetry axis of symmetry that is perpendicular to our directrix and passes through our focus so it's here this is our axis of semen [Music] so x is equal to negative one and last the end points of the lattus rectum so endpoints of lattice rectum we have the formula 2c and our c is 2 so substitute 2 times our c is 2. so this is four so from the focus four units to the right one two three four so our coordinates is nega positive 3 then 1 2 3 4 5 5 and the other end point focus 5 units now four units one two three four four units so we have the coordinates negative one negative two negative three negative four negative five two positive five for our y-axis okay the end points are the right-most point is three five and the left-most is negative five five now connect our parabola so vertex connect now this is now our parabola so the length of lattice rectum land of lattice vector is one two three four five six seven eight eight or simply four c four times two that is eight units okay we're done and this is now our graph transforming general form the standard form of the parabola this is the continuation of our lesson in our videos parabola if you haven't watched the video please go to our playlist or just click the card so it will direct you in that video the general form of the parabola is y is equal to a x squared plus b x plus z while the standard form is quantity x minus h raised to 2 is equal to 4p quantity y minus k we're going to transform into a standard form let's say we have this given x squared plus 10x minus 2y plus 23 is equal to zero we're going to write into standard form i'll transform into standard form so let's begin this copy first given so x squared plus 10 x minus 2y plus 23 is equal to zero it's just like we're we're just solving uh uh what they call this rubik's cube or a positive so we need to do we have the square of a binomial on our left side so our x x squared plus 10 x then move other expressions or other terms on the right side will become 2y minus 23 positive to y move the left that will become positive and negative 2y will become positive to y and negative positive 23 will become negative 23. now we have a square of a binomial that means we need to make our left side of the equation a perfect square trinomial so x squared plus 10 x plus blank is equal to 2y minus 23. we added black on the left side we need to add blank on the right side to make our equation balance now what would be the value of the plan i completed the square in the middle term 10 divided by 2 that is 5 5 square 25 25 and 25 now this is a perfect square trinomial so writing square of a binomial so the square root of x squared that is x copy the sign of the middle term steve the square root of 25 is five then squared is equal to simplify 2y negative 23 plus 25 capacity so common factor copy x plus 5 squared is equal to common factor is two now we have y plus one this is now our standard form now your turn i want you to try this problem 5y squared plus 30y plus 24 is equal to 51. 5y squared plus 30y plus 24 is equal to 24x is equal to 51. right into standard form so in this case we have y squared so our y squared must be a perfect square trinomial so in our standard form it depends well on the openings of the parabola so go to our first video so if you want to try pause the video then try to answer this problem then after answering resume watching to check your answers here's our solution so 5 y squared plus 30y is equal to negative 24x plus 51. so first divide the equation by five so this will become y squared plus 30 divided by 5 6 y is equal to negative 24 over 5 x plus 51 over 5. so that we can make our right side a perfect square trinomial now y squared plus 6 y plus 1 is equal to negative 24 over 5 x plus 51 over five we added blank so we need to add the last on the right side now get the third term middle term six divided by two that is three three square nine added nine on the left add nine on the right side okay this is now a perfect square trinomial right into square of a binomial that is y plus square root of nine three squared is equal to combine negative 24 over 5x let's combine lcd is 5 5 divided by 5 that is 1 times 51 so 51 plus 5 divided by 1 that is 5 times 9 45 now we have y plus 3 squared is equal to negative 24 over 5x add so 51 plus 45 that is 96 over 5. and now get the common factor so copy y plus three raised to 2 is equal to the column factor is negative 24 over 5. we have now x plus 96 divided by negative 24. now since we have negative here this will become negative because negative times negative will give us positive 96 divided by 24 that is four over over one or simply four this is now our standard that's on how to convert general form to our standard form so after discussing is a general form to standard form now let's have transforming standard form to general form of the parabola let's have letter a b and c as our problem for letter a quantity y minus 3 raised to 2 is equal to negative 2 quantity x plus 1. we're going to write it into general form let's start so square of a binomial we need to expand so square the first term that will be y squared multiply the first term and the second term times two so y times negative three negative three y times two negative six y square the second term positive nine is equal to distribute negative two times x negative two x negative two times one negative okay now equate to zero so we have y squared minus six y plus nine plus two x plus two is equal to zero then rare range and combine like terms so we have y squared plus two x minus six y nine plus two that is positive eleven is equal to zero that will be our general form letter b so just copy y squared then distribute is equal to three times x that is three x three times negative one negative three now equate to zero so we have y squared minus three x plus three is equal to zero this will be our general form and our last example quantity y plus 3 raised to 2 is equal to negative 24 over 5 quantity x minus 4. so we have a fraction five okay let us expand first our quantity y plus three that is to two so square the first term that is y squared multiply the first term and the second term that is three y times two so positive six one square the second term positive nine is equal to distribute negative 24 x over five and negative times negative pass it to you 24 times four let us multiply 4 times 4 16 01 4 times 2 8 plus 1 9 so that is 96 over 5. next equate to zero so we have y squared plus 6y plus 9 plus 24 x over 5 minus 96 over 5 is equal to zero okay we can remove five we're going to multiply the equation by five so five times y squared 5 y squared 5 times 6y positive 30y 5 times 9 positive 45 so that will be cancelled out plus 24x minus cancel out 96 is equal to 5 times 0 0 okay rearrange 5y squared plus 30y plus 24x and now let us combine 45 minus 40 out 45 minus 96 so that is negative so 96 minus 45 1 5 so negative 51 is equal to zero this will be our general one so in order for you to master this is standard for the general form notice that we have a square of a binomial so you can go to our grade 8 playlist then answer and then watch our lesson one spatial products for your assignment you can answer the following problem okay assignment i will raise b and i will write your assignment here right now transform the standard form of the equation of the parabola to its general form so number one y minus eight raised to two is equal to negative three x plus one letter b x squared is equal to three times y plus four raised to three oh no that's it and letter c x plus 4 raised to 2 is equal to 3 times y minus 1. this will be your assignment transforming standard form to the general form of the parabola your next lesson will be the ellipse so before watching your next lesson you need to master the circle and of course the parabola because you need these steps in parabola and circle in our lesson daily once again thank you for watching senior pablo tv don't forget to subscribe click the notification bell in order for you to be updated in our upcoming videos and of course share this video to your classmates to help them answering their [Applause] modules