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Int Algebra: Writing an Equation for a Parabola


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About this Lesson

  • Type: Video Tutorial
  • Length: 5:08
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 55 MB
  • Posted: 12/02/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Trigonometry: Full Course (152 lessons, $148.50)
Intermediate Algebra Review (25 lessons, $49.50)
Trigonometry: Conic Sections (12 lessons, $26.73)
Trigonometry: Parabolas (4 lessons, $5.94)

A parabola is a conic section in which the locus of points constructing it are equidistant from the focus and the directrix. To find the formula of this equation when given the vertex (h,k) and the distance from the focus (p), this lesson will show you how to find the equations for the parabola described by these criteria. There will be two formulas depending on whether p is positive or p is negative (which should indicate whether the parabola opens up or down).

This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Intermediate Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers real numbers, equations and inequalities, exponents and polynomials, rational expressions, roots and radicals, relations and functions, the straight line, systems of equations, quadratic equations and quadratic inequalities, conic sections, inverse and exponential and logarithmic functions, and a variety of other AP algebra and advanced algebra.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit or visit Thinkwell's Video Lesson Store at

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Recent Reviews

Great lesson
~ hmt79

This lesson does a good job of showing you how to figure out the equation for a parabola provided you have some pieces of information but lack the equation itself - very helpful and very easy to understand.

Great lesson
~ hmt79

This lesson does a good job of showing you how to figure out the equation for a parabola provided you have some pieces of information but lack the equation itself - very helpful and very easy to understand.

Conic Sections
Equations for Parabolas
Writing an Equation for a Parabola Page [1 of 1]
Suppose you actually want to find the equation of a parabola, I don’t know why, and all you have at your disposal is
the vertex. Remember that’s sort of the critical point where the parabola turns. And someone also tells you the
distance that point is from the focus. How would you proceed? Well, if you have a vertex at (h, k) and your focus is a
distance either plus or minus p away, depending on whether p is negative or not. Then here’s the poop. Either the
equation is (x - h)² = 4p(y - k) and that’s if you have parabola that’s either going to be a happy face or a sad face.
Depending upon whether p is positive or p is negative.
The other possibility is that you have (y - k)² = 4p(x - h). In which case then you have either a opening parabola this
way, if p is positive, or a parabola that opens this way if p were negative. So, in fact, that’s the whole story. You can
easily find the formula for anything that someone gives you. For example, let’s suppose I tell you, “I’m thinking of a
parabola and its vertex is at (0, 0) and its focus is (4, 0).” Name that parabola. What would you do?
Well, first of all I would sketch a picture of it to see what’s going on here. So, if you sketch a little picture of it, it would
look something like this. Well, let’s see. We’ve got the vertex here at (0, 0) and the focus is at (4, 0). So, if that’s the
focus the parabola sort of wants to go around the focus. So, in fact, the parabola must somehow look like this. So,
that tells me it must be of this form and p must be positive. So, that’s pretty good. So, I know it has to look like this.
So, I know what the vertex is. It’s (0, 0), so I put a 0 in for k. I put a 0 in for h. That’s 0. So, I see just y² equals and
now what’s p? Well, p is the distance away from the focus. What’s the distance the vertex is from the focus? It’s
four. So, I put a 4 for p. So, 4 x 4 would be 16x. So, that’s the equation of the parabola that has vertex (0, 0) and
focus at (4, 0). Neat and pretty easy too.
Let’s try another one together. Suppose I tell you I’m thinking of a parabola and the vertex of this parabola is located
at (4, 3) and its focus is located at (4, 0). Name that parabola. The first thing I would do is sketch a picture of it and
see what this thing looks like. So, what have we got going on here? The vertex is at (4, 3). So, there’s the vertex and
the focus is at (4, 0). So, there is the focus and the parabola wants to curve around the focus. So, the parabola looks
like this. Which means it’s a parabola that goes up or down. So, it’s like this equation and it’s a sad face parabola.
So, p is now going to be negative.
So, let’s see if we can now report the answer. What would we have? I would see that I’d have--here’s the vertex. So,
what would I see? I would see x - h. Now, the h here is 4 and the k here is 3, because that’s the vertex. So, I’d see
(x - 4)² equals and now what’s p? Well, p here I remind you, is suppose to be negative, because I want it to be a sad
faced parabola and it represents plus or minus the distance away the vertex is from the focus. So, what’s this
distance? This is at three. This is at zero. So, the distance is three. So, it’s three, but I want it to be negative
because I want it to be a sad face parabola. So, -3 x 4 is -12 times y and then I have -k. What’s k? Well, k is three.
So, I have minus three.
So, that is the equation for the unique parabola that has a vertex at (4, 3) and a focus at (4, 0). Weren’t you always
wondering how to find the equation of that parabola? Now you know, and if you actually can use this good for you.
See you soon.

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