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Lesson 9:
Deriving the Vertex Form of the Equation of the Parabola
Sasha and Keoni develop the vertex form of the equation of a parabola as y = (x–h)2/(4p) + kwhere the (h,k) is the vertex and p the distance from the vertex to the focus.
Episode 1: Making Sense
Sasha and Keoni examine the different equations they derived for parabolas with a p-value of 3 and a vertex not at the origin. By noticing patterns between the location of the vertex and the equation for the parabola, they make a prediction for the general equation of a parabola with a vertex at (h, k) and an unknown p-value.
Episode 2: Exploring
Keoni and Sasha use the Pythagorean Theorem and the definition of a parabola to derive an equation of parabola with a p-value of 5 and a vertex at (9, 13).
Episode 3: Making Sense
Keoni and Sasha use the applet to explore the graphs of parabolas with a vertex at (9, 13) and an unknown p-value. Sasha and Keoni determine how to represent the coordinates of the focus and the equation of the directrix when p can take on any value.
Episode 4: Exploring
Sasha and Keoni extend their work from the last episode to derive the equation of a family of parabolas that have a vertex at (9, 13).
Episode 5: Reflecting
Sasha and Keoni look back on their work with a parabola with a p-value of 5 and a vertex of (9, 13). By reflecting on their work and the equation y = (x – 9)2/(4p) + 13, they see 13 their graph.
Episode 6: Making Sense
Sasha and Keoni build on what they have learned in the previous episodes to begin to develop the general equation for any parabola with vertex (h, k) and the distance p from the vertex to the focus.
Episode 7: Exploring
In the last episode Sasha and Keoni determined the distances of the sides of a right triangle on a general parabola with vertex (h, k) and distance p from the vertex to the focus. Next they derive the equation for the parabola by substituting those distances into the Pythagorean Theorem.