# Parabolas Lesson 5

### 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.

##### Episode 1: Making Sense

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.

##### Episode 2: Exploring

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.

##### Episode 3: Reflecting

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.

##### Episode 4: Repeating Your Reasoning

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.

##### Episode 5: Making Sense

Sasha and Keoni make a prediction about an equation for any parabola with a vertex on the origin.

##### Episode 6: Exploring

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.

##### Episode 7: Reflecting

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).