# Parabolas Unit (Teachers)

Students work from the geometric definition to construct a parabola using the focus and directrix. They use the Pythagorean theorem to connect the geometric definition to the algebraic representation of a parabola on the coordinate grid. Through a series of activities, they develop the vertex form of the equation of a parabola. They also learn to identify geometric information, such as the coordinates of the focus and vertex, from the various forms of the equation of a parabola.

##### Lesson 1: Creating a Parabola from the Geometric Definition

Sasha and Keoni create a parabola from its geometric definition. This involves making sense of key terms—like focus, directrix, and equidistant—and figuring out how to measure distances between points and lines.

##### Lesson 2: Connecting Geometry with Algebra

Students work from the geometric definition to construct a parabola using the focus and directrix. They use the Pythagorean theorem to connect the geometric definition to the algebraic representation of a parabola on the coordinate grid. Through a series of activities, they develop the vertex form of the equation of a parabola. They also learn to identify geometric information, such as the coordinates of the focus and vertex, from the various forms of the equation of a parabola.

##### Lesson 3: Developing an Equation for a Parabola for Any Given y-Value

Keoni and Sasha create a general method for representing the x-value for any point on a particular parabola, given the y-value of that point. By using their previous results, along with the Pythagorean theorem, they are able to determine the equation for the parabola.

##### Lesson 4: Developing an Equation for a Parabola Given Any x-Value

Sasha and Keoni use the definition of a parabola, the Pythagorean theorem, and their methods from previous lessons to represent the y-value for any point on a particular parabola given the x-value of that point. In contrast to Lesson 3 in which y-values were given, in this episode x-values will be given.

##### 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.

##### Lesson 6: Exploring a Parameter Change

Keoni and Sasha compare the graphs of y = x2/(4p) for p-values of 1/4, 1/2, and 1. They figure out the effect that changing the value of p has on the graph of the parabola.

##### Lesson 7: Explaining a Parameter Change

Sasha and Keoni use algebraic and geometric thinking to form three arguments that justify why a parabola gets wider on the coordinate grid as the p-value in y = x2/(4p) increases.

##### Lesson 8: Exploring Parabolas with Vertex (h, k)

Sasha and Keoni use a GeoGebra applet to move parabolas to the left, right, up, and down. Then they develop equations for several different parabolas where the vertex is not at the origin.

##### Lesson 9: Deriving the Vertex Form of the Equation of a Parabola

Sasha and Keoni develop the vertex form of the equation of a parabola as y = (x–h)2/(4p) + k where the (h,k) is the vertex and p the distance from the vertex to the focus.

##### Lesson 10: Getting and Using Geometric Information

Given the equation of a parabola in any form, Sasha and Keoni find geometric information (such as the focus, directrix, p-value, and vertex) about the parabola.