Finding the standard form of a parabola given vertex and focus

Learn how to write the equation of a parabola given the vertex and the focus. 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 vertex of a parabola, recall that the focus of a parabola 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

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[motheroftwo]

Not sure if this will help, but many of you asked about what to do if the focus has a fraction, as my problem does. What he did without really saying it is he took the 4 from the focus and subtracted the 1 from the vertex to get 3. So, for instance if you have vertex (3, 5) and focus (3, 79/16), you would do this: 79/16 minus 5/1 and find the least common denominator to get -1/16 as your "p". Once you do this, just slip it into your formula: (x-3) ^2 = 4 (-1/16) (y-5). Of course, you will need to multiply 4/1 and -1/16 to get -1/4. And then just use all your algebra 2 mathematics to solve the problem. Your final answer will be y = -4(x-3)^2 +5. I hope this is not confusing, if so I can give more details.

[jigsawpuzzle]

Im new to your subscription 😊 Thanks alot now I know how to find the standard equation 😊... So sir is it the final answer? Or I need to multiply it? Cause mine is X=1/8(x² - 4x +12)

[mustafamalik]

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[ghostbusterkane]

With online learning this really helps!! thank you!

[sourabhpandey]

I am from india . It clears my concept.thnx dear professor

[marfeliz]

Another way that can help you figure out which standard equation to use is by remembering that the (x-h)^2=4c(y-k) equation is to look for the number in the coordinate that the vertex and focus both have in common. In this case that number is two, which lies on the X portion of the coordinate. Thereby, I remember that I have to use the (x-h)^2=4c(y-k) formula. If the vertex were at (0, 3) and focus (0, -1) then I would find that the 0 is the common number that they both share, which is on the x and write the equation for the origin : X^2=4cy.

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[khushisingh]

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You helped me a lot.