Writing the equation of a parabola given focus and directrix

Learn how to write the equation of a parabola given the focus and the directrix. A parabola is the shape of the graph of a quadratic equation. A parabola can open up or down (if x is squared) or open left or right (if y is squared). Recall that the focus and the vertex of a parabola are on the same line of symmetry.
When given the focus and the directrix of a parabola, recall that the vertex of a parabola is halfway between the focus and the directrix and the focus is inside the parabola. This enables us to identify the direction which the required parabola opens. We also need to identify the value of p, which is the distance between the vertex and the focus. p is negative when the parabola opens down or left and is positive when the parabola opens right or up.
Once we identify the direction and the value of p, we can use the equation of parabola given by (y - k)^2 = 4p(x - h) for parabolas that opens up or down and (x - h)^2 = 4p(y - k) for parabolas that opens left or right.
#conicsections #parabolaconicsections

[@pancakes246]

These students don't know how lucky they are.

[@vyntonix7722]

THIS MAN STILL COMING IN CLUTCH 4 YEARS LATER

[@fawazza]

so much better than my algebra 2 teacher

[@brianmclogan]

happy to be able to assist you!

[@stutch4048]

Yep, same here. Thanks for making it easy to understand

[@kaskadelynn554]

*De focus is a point, so we plot it as a point. De directrix is a line, so we plot it as a line.* Someone should make a remix of this.

[@brianmclogan]

haha

[@deadmemes7554]

I can’t disagree

[@jedansoringa9549]

I'm not literally a fast learner first I didn't understand all of what you say but I repeat and repeat as I want to repeat. Then, thankfully I understand it so much... Thank you 😍☺☺☺

[@marfeliz7143]

SIR YOU ARE A BRAIN! I am in college right now and I am going to take my exam 4 this week. I was freaking out but I thank God he made you a professor because you just served your purpose in life, TO HELP PEOPLE LIKE ME!!! I love your videos! YOU AMAZING BALD BRAIN!!!!!! love u

[@brianmclogan]

haha cheers Mar. Best of luck on your exam

[@twentyfivehourpharmacy6348]

I’m taking this as a sophomore in HIGHSCHOOL. I’m so done with getting the same damn math as people in college 😭

[@peachyfire4904]

i’m learning it in 8th grade and it’s so hardddd

[@peachyfire4904]

@@twentyfivehourpharmacy6348SAME

[@peachyfire4904]

@@twentyfivehourpharmacy6348DID U PASS THE TEST ON THIS UNIT? MINE IS TOMORROW

[@emptytears47a]

This guy has a lot of great info. I have been struggling with math all summer with the videos my class provided and I look at one of his videos and he just simplified this for me where I know it now, thank you

[@qedmath1729]

This is how i like to do it. We know the eccentricity of a parabola is 1. Meaning the distance from a point (x,y) to the focus is the same as that point to the directrix. So simply write the line x=5 as x-5=0. And use the distance from point to line formula to find the distance from a point (x,y) to x-5=0. Next note this is equal to the distance from point (x,y) to (2,-3). Setting up the equality, squaring both sides and rearranging terms will give you equation of the parabola.

[@robotking5454]

Sir Can you explain how you got the 3.5 and -3 and why the formula is not (y-h)squared first

[@mossfeathers]

any other people here at midnight desperately trying to find out how to do this?

[@jrseahorse5886]

My teacher introduced us to geometric properties of a parabola 2 days ago and today I have a test on it. I’m pretty sure I’m gonna fail but I will still study.

[@jahzielperalta484]

SAME

[@xuzixi1389]

SAME

[@lizalonso_]

wait where did the 1.5 come from ?

[@SleazySnailz]

bruh idk either

[@juandiegomoreno9803]

late reply but for anyone who sees this and is confused... well here's my best way to explain it: that point, which is 1.5 units away from the directrix and the focus, is where the parabola is. the parabola will always be equally as far from the directrix as the focus, so that means it's at half. Therefore, if the distance from the focus to the directrix is 3, then the parabola will be located at 1.5 points between the focus and the directrix, in this case being coordinates (3.5, -3)

[@jaymafiaa8915]

how did you know to go over 1/2 spaces?

[@ryanrollins686]

Bc the vertex is halfway in between the focus and directrix

[@arefasuleyman2379]

I like the way he smiles at end

[@nisah4179]

WOW your videos really helped me a lot in understanding our next topics in class. Thank you!

[@brianmclogan]

you are welcome!

[@Nobody-tj9jo]

How do I know which way a parabola opens when given the focus and directrix?

[@luzl.1403]

Oh my god thank you so much you simplified this and I understood this in literally 4 min!!

[@syntaxerror2409]

so if your parabola faces to the right the so you will still use (y-k)^2=4c(x-h) not (y-k)^2=-4c(x-h)?

[@brianmclogan]

that is correct, 4 is always positive, if the parabola opens to the right, c is positive, if it opens to the left c is negative

[@faithannwood1257]

You might just be saving my life. are there any videos on this but using the process of Deriving the formula by equating the distances? because that is how i'm being told to do it :(

[@theturnmusic9164]

dope,all the way from Namibia africa learning from a distance.

[@irislangen3925]

you dont even know how much this video helped me

[@Ursobeautifuleveryday]

Where did you find 1.5 . And can you show how to find vertex algebraically. I dontunderstand howdid youcalculate 3.5

[@serandure8163]

Thank you so much for this video it really heloed me with my math test

[@meaghansnider1065]

So how come you use the vertical equation when the parabola is horizontal?

[@korzinietz2136]

It was really helpful. Thank you very much.

[@brianmclogan]

You are very welcome!!

[@nickschultz6611]

Your channel is a lifesaver. Thanks !

[@tinyade.s]

Thank you so much this helped for my math test today!!

[@adrisoa]

THANK UUUUUUUUUUU. I HAVE A TEST IN 30 MINS AND U MADE ME UNDERSTAND WHAT TO DO!

[@brianmclogan]

Awesome so happy to help!!

[@jeramiahcollins2895]

bruh my math teacher was literally speedrunning through this and this makes so much more sense

[@alexiajaniece9190]

My favorite teacher and I don’t even have you🙂

[@brianmclogan]

happy to help you out!

[@swrrsssooowws126]

My Algebra 2 teacher is good I just forgot lol

[@isaaclivai]

Thanks alot man, your tutorials are very transparent and easy to understand.. With your help i really enjoyed math thanks

[@gemaramirez6903]

Thank you for helping 😩

[@shyalanar82]

You're a wizard, Brian

[@eliafora4403]

Hes literally the only math teacher on youtube that makes sense to me

[@derekmolina9254]

Why is it 4p

[@brianmclogan]

that is the formula, there are proofs on it if actually looking for the mathematical approach

[@junhaosu373]

This helped so much.

[@user-wd9pd9mv7g]

Sir, how do you find 3.5?

[@felipemoraleja5280]

Equation for vertex I came up since it doesn't tell you: FOCUS (h, k) directrix = x x - h = d d/2 = m h + m = v (vertex 'x') 'y' is the same as the original 'k' in (h, k) (v, k) is the vertex

[@hxnnas_vs]

Thank you Felipe

[@sashat3999]

You’re godsent

[@marielsabanal7137]

thank you for teaching us

[@jeremycanizares3594]

I wish I could participate in one of his classes lol

[@jiyasgiantsongs3554]

I literally love this man

[@jschampion1814]

drawing graph is must cant i just find the focus and vertex without drawing the graph...

[@brianmclogan]

sure thing, I just do it because It helps

[@nickmix_5809]

you are amazing

[@amberskye93]

my tutor was explaining it wrong to me on zoom while I was playing this in the background and explained it correctly to them lmao maybe they did that on purpose? As always thanks for coming in clutch!

[@omarolad4671]

Sir how can i find the vertex like this Example without graphing it ? Please answer.

[@joeldoxtator9804]

NO WHERE in my textbook did it mention the focus and the directrix are an equal distance from the vertex! I can see that now looking at the equation of both, but god damned an explanation of that would have been helpful instead of just throwing a problem at you with only focus and directrix without explaining that!

[@amgie5958]

Thank you so much!

[@ye4045]

Thank You so much

[@JC-po2cv]

What Algebra 2? I'm in college pre calc and trig and I had this on my exam today and felt so dumb..... Thank you though.

[@brianmclogan]

I teach this in pre-calc as well, so no worries

[@alexdig5761]

I appreciate you so much sir, you don’t even understand

[@dylanjohansen7360]

man where can I get all the stuff on the right of the board

[@brianmclogan]

That $$ stuff

[@stephanielopez3320]

wow im in 9th grade about to start taking 10th grade classes so im watching this video and they are really helping me

[@brianmclogan]

awesome happy to help

[@worldmonitor5132]

How does x in vertex becomes 3.5 ...if anyone can explain.

[@genosect1744]

How to find the parabola given only its focus?

[@meenalgautam1465]

is it 1.5 or 3.5

[@poboyplayz9909]

wish my teacher was like this

[@muhammadyahya8157]

Sir how to solve this one F(-3,1) and directrix x-2y-3=0 this one is diffrent

[@brianmclogan]

Sorry I am only able to focus on what I am currently teaching. I do have a playlist with multiple examples

[@muhammadyahya8157]

Brian McLogan ohh ok sir..im just stuck with this one i have checked all the playlist but could nt find one like it..

[@brianmclogan]

sorry, try mathway.com you can at least check the answer for free

[@floaxclive6042]

no square on both (x or y), i suppose it's not even a valid parabola equation. prove me if I'm wrong though(too lazy to look it up...) Edit: ohhh directrix..... sorry

[@nikkydelacruz113]

so to find the vertex,it is the distance of focus to directrix,am i correct sir?

[@brianmclogan]

no it is perpendicular to the directrix and equal distance between the two

[@Yzurra]

To make it more clear, you can find the distance between the focus to the directrix and divide the number by 2 to find the value of one of the coordinates of the vertex. For example: Focus(3,4) Directrix: y=8. You find the distance between the vertex and the directrix. The directrix is y=(#), so you can forget about the x value of the focus and work with the y values for now and find the distance between the focus and the directrix. In this case, it is 4, and then you can divide that number by 2(you get 2) to find the distance between the vertex and the focus(this works because the vertex is exactly the same units from the directrix and the focus(which means in between)). So you get 2 by dividing 2. You can add that to the y value of the focus or subtract it depending on which way the vertex is facing. In this example, the directrix is above the focus, and vertex always faces the on the focus, so it will be adding since the vertex would also be above the vertex, facing downwards(let that sink in). By adding the 2 to the y value of the focus, you get (3,6) as your vertex. TL;DR If the directrix is *x* =(#), add that number to the *x* value of the Focus and divide by 2. The y value stays the same. And vice versa. P.S. Please point out any mistakes I've made so I can make corrections.

[@manisinterlude]

@@Yzurra Omg you’re so smart, thanks

[@nanamikents1760]

thank you!!!

[@regineshrinebalgos5651]

Where is the 3.5 came? Even though the distance is 3?

[@brianmclogan]

the distance between the focus and directrix is 7 and the vertex is in the middle

[@junaidmughal3806]

I see your videos I press thumbs up

[@abelhagos3142]

What grade lesson is this in your country???

[@randomsprinkles7159]

11th ish

[@Mulla-alyusufi]

🎉🎉🎉

[@ophyjr2611]

I almost killed myself and you saved me thank you

[@alyzamorales7333]

2020 and this helps, my algebra teacher is lameeee

[@kaitlynna.7869]

how did he get 4.... i'm so stupid

[@elirose6033]

My teacher taught this yesterday and now we have a test >_> ima fail

[@jrseahorse5886]

Elizabeth Lance same we got introduced to it 2 days ago and now I have a test

[@esharana3818]

How do we know if directrix is vertical or horizontal?

[@brianmclogan]

if x is squared then the directrix is horizontal, if y is squared then the directrix is vertical

[@teolapad485]

Nice opaw😅

[@hihi-vk5gg]

i’m here bc ive been crying over my simple alg 2 hw for 2 hours i literally can’t do shit

[@hihi-vk5gg]

ok i got it. ur literally epic it feels like a weight has been lifted off my shoulders lol

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bro me too

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[@maxmeadowcroft1805]

no

[@brianmclogan]

Okay