Course: College Algebra
Topic: Conic Sections
Subtopic: Parabolas


One of the conic section curves is the parabola. In our previous study of parabolas (graphs of quadratic equations) the equations were generally solved for y, e.g., `y=x^2-3x-4` or `y=(x-3/2)^2-25/4` (*). But in today's lesson we learn a different form of the equation of a parabola, one that is solved for the square term, e.g., `x^2=2y-5`. This form actually gives more information and is the form used most often in calculus. This new approach allows us to graph parabolic functions (those that open up/down) as well as sideways parabolas (ones that open left/right and are non-functions).

* Try me: Both of these parabolas open up from the vertex point `(3/2,25/4)`. Can you complete the square to transform the first equation to the second? Can you verify this vertex from both forms of the equation?


By the end of this topic you should know and be prepared to be tested on:


Terms you should be able to define: parabola, parabolic function, sideways parabola, general form (e.g. `y=ax^2+bx+c`), standard form (e.g. `(x-h)^2=p(y-k)^2`), vertex, axis of symmetry, focal point, directrix line, latus rectum

Supplemental Resources (recommended)

Explore the interactive graphs at Interactive Parabola Graphs which show how the parabola's focal point and directrix line work together to form the parabola's unique shape. The Parabola has a couple nice applications of parabolas, worth skimming.

Supplemental Resources (optional)

If you need supplemental tutorial videos with examples relevant to this section go to James Sousa's MathIsPower4U and search for topics: "Graphing and Writing Equations of Parabolas".

rev. 2020-11-29