Deriving an Equation for all Parabolas with Vertex at the Origin
Keoni and Sasha determine a general method for representing the equation for a parabola with a vertex on the origin and a focus p units above the vertex.
Sasha and Keoni examine the similarities and differences between two parabolas with vertices on the origin but with different distances to the focus. They determine if the same equation could represent both parabolas.
Keoni and Sasha develop an equation for a parabola with a focus at (0,2) and a directrix of y = –2. They use the Pythagorean theorem and the definition of a parabola.
Keoni and Sasha compare the equations for two parabolas: y = x2/8 and y = x2/4. They make two different conjectures about the equation for a parabola with a focus at 3 units above the vertex.
Keoni and Sasha determine the equation of a parabola with a vertex at (0,0) and distance of three units between its focus and vertex. They compare the equation to their conjectures from Episode 3.
Sasha and Keoni use the Pythagorean theorem and the definition of a parabola to derive the equation for a parabola with a vertex at the origin and a distance of p between the focus and vertex.
Sasha and Keoni discuss what the equation y = x2/(4p) means. They also use it to find the equation of a parabola with a vertex at the origin and a focus at (0, 0.5).
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
1. Interpret the meaning and use of an equation that they derive to represent a family of parabolas with vertex at the origin and focus p units above the vertex (y = x2/(4p)).
2. Conceive of the parameter p in the equation y = x2/(4p) as a distance between the vertex and focus of a parabola. The value of p can vary, but once it is set (e.g., when p = 2), then an equation is defined (e.g., y = x2/8) that represents a unique parabola.
3. Relate geometric features of parabolas to elements of corresponding equations. For example, the equation y = x2/8 can also be expressed as y = x2/(2•4), where the 2 represents the number of units between the focus and vertex of the parabola.
Common Core Math Standards
CCSS.M.HSG.GPE.A.2. Derive the equation of a parabola given a focus and directrix.
In Episode 6, Sasha and Keoni derive the equation (in two forms) for the family of parabolas with vertex (0,0) and focus p units above the vertex, namely y = x2/(4p) and x = √(4py). They do this by generalizing the method they used to develop an equation for a parabola with p = 1 in Lessons 3 and 4 (y = x2/4 and x = √(4y)), for a parabola with p = 2 in Episode 2 of this lesson (y = x2/8 and x = √(8y)), and for a parabola with p = 3 in Episode 4 (y = x2/12 and x = √(12y)).
Sasha and Keoni multiply binomials, such as (y + 2)2 and (y + p)2, in service of deriving an equation for a particular parabola (in Episodes 2 and 4) or a family of parabolas (Episode 6).
Keoni and Sasha re-express the equation for a given parabola that they derived in Lesson 3 (namely, x = √(4y)) as y = x2/4. They discuss how the first form of the equation is easier to use when one is given a y-value and needs to determine the corresponding x-value, while the second form is easier to use when one is given an x-value. However, they recognize that the two forms are equivalent and that either can be used to solve both problems.
In this lesson, Sasha and Keoni make use of mathematical structure on two levels. First, they discern a pattern relating the equations of three parabolas (y = x2/4, y = x2/8, and y = x2/12) with the corresponding distance between the vertex and focus (respectively, p = 1; p = 2; and p = 3). Specifically, they see the structure of 1•4, 2•4 and 3•4 in the denominators of their equations, which is the p-value of the parabola multiplied by 4. Consequently Keoni and Sasha conjecture that the general equation of a parabola with vertex (0,0) and focus p units above the vertex will be y = x2/(4p). Then they look for and make use of structure on a second level. To derive the equation y = x2/(4p), they need to identify the lengths of the sides of a right triangle with hypotenuse connecting a general point on the parabola with the focus. To accomplish this challenging task, Sasha and Keoni see a pattern in similar quantities that they identified for particular parabolas, which they generalize to define lengths involving the parameter p, such as y – p and y + p.
