Sasha and Keoni create a parabola from its geometric definition. This involves making sense of key terms\u2014like focus, directrix, and equidistant\u2014and figuring out how to measure distances between points and lines.<\/p>\n\n\n\n
Lesson 2: Connecting Geometry with Algebra<\/a><\/h5>\n\n\n\nStudents 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.<\/p>\n\n\n\n
Lesson 3: Developing an Equation for a Parabola for Any Given y-Value<\/a><\/h5>\n\n\n\nKeoni 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.<\/p>\n\n\n\n
Lesson 4: Developing an Equation for a Parabola Given Any x-Value<\/a><\/h5>\n\n\n\nSasha 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.<\/p>\n\n\n\n
Lesson 5: Deriving an Equation for all Parabolas with Vertex at the Origin<\/a><\/h5>\n\n\n\nKeoni 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.<\/p>\n\n\n\n
Lesson 6: Exploring a Parameter Change<\/a><\/h5>\n\n\n\nKeoni and Sasha compare the graphs of y = x2<\/sup>\/(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.<\/p>\n\n\n\nLesson 7: Explaining a Parameter Change<\/a><\/h5>\n\n\n\nSasha 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<\/sup>\/(4p) increases.<\/p>\n\n\n\nLesson 8: Exploring Parabolas with Vertex (h, k)<\/a><\/h5>\n\n\n\nSasha 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.<\/p>\n\n\n\n
Lesson 9: Deriving the Vertex Form of the Equation of a Parabola<\/a><\/h5>\n\n\n\nSasha and Keoni develop the vertex form of the equation of a parabola as y = (x\u2013h)2<\/sup>\/(4p) + k where the (h,k) is the vertex and p the distance from the vertex to the focus.<\/p>\n\n\n\nLesson 10: Getting and Using Geometric Information<\/a><\/h5>\n\n\n\nGiven 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.<\/p>\n\n\n\n
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.<\/p>\n\n\n\n
Lesson 4: Developing an Equation for a Parabola Given Any x-Value<\/a><\/h5>\n\n\n\nSasha 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.<\/p>\n\n\n\n
Lesson 5: Deriving an Equation for all Parabolas with Vertex at the Origin<\/a><\/h5>\n\n\n\nKeoni 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.<\/p>\n\n\n\n
Lesson 6: Exploring a Parameter Change<\/a><\/h5>\n\n\n\nKeoni and Sasha compare the graphs of y = x2<\/sup>\/(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.<\/p>\n\n\n\nLesson 7: Explaining a Parameter Change<\/a><\/h5>\n\n\n\nSasha 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<\/sup>\/(4p) increases.<\/p>\n\n\n\nLesson 8: Exploring Parabolas with Vertex (h, k)<\/a><\/h5>\n\n\n\nSasha 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.<\/p>\n\n\n\n
Lesson 9: Deriving the Vertex Form of the Equation of a Parabola<\/a><\/h5>\n\n\n\nSasha and Keoni develop the vertex form of the equation of a parabola as y = (x\u2013h)2<\/sup>\/(4p) + k where the (h,k) is the vertex and p the distance from the vertex to the focus.<\/p>\n\n\n\nLesson 10: Getting and Using Geometric Information<\/a><\/h5>\n\n\n\nGiven 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.<\/p>\n\n\n\n
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.<\/p>\n\n\n\n
Lesson 6: Exploring a Parameter Change<\/a><\/h5>\n\n\n\nKeoni and Sasha compare the graphs of y = x2<\/sup>\/(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.<\/p>\n\n\n\nLesson 7: Explaining a Parameter Change<\/a><\/h5>\n\n\n\nSasha 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<\/sup>\/(4p) increases.<\/p>\n\n\n\nLesson 8: Exploring Parabolas with Vertex (h, k)<\/a><\/h5>\n\n\n\nSasha 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.<\/p>\n\n\n\n
Lesson 9: Deriving the Vertex Form of the Equation of a Parabola<\/a><\/h5>\n\n\n\nSasha and Keoni develop the vertex form of the equation of a parabola as y = (x\u2013h)2<\/sup>\/(4p) + k where the (h,k) is the vertex and p the distance from the vertex to the focus.<\/p>\n\n\n\nLesson 10: Getting and Using Geometric Information<\/a><\/h5>\n\n\n\nGiven 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.<\/p>\n\n\n\n
Lesson 7: Explaining a Parameter Change<\/a><\/h5>\n\n\n\nSasha 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<\/sup>\/(4p) increases.<\/p>\n\n\n\nLesson 8: Exploring Parabolas with Vertex (h, k)<\/a><\/h5>\n\n\n\nSasha 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.<\/p>\n\n\n\n
Lesson 9: Deriving the Vertex Form of the Equation of a Parabola<\/a><\/h5>\n\n\n\nSasha and Keoni develop the vertex form of the equation of a parabola as y = (x\u2013h)2<\/sup>\/(4p) + k where the (h,k) is the vertex and p the distance from the vertex to the focus.<\/p>\n\n\n\nLesson 10: Getting and Using Geometric Information<\/a><\/h5>\n\n\n\nGiven 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.<\/p>\n\n\n\n
Lesson 8: Exploring Parabolas with Vertex (h, k)<\/a><\/h5>\n\n\n\nSasha 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.<\/p>\n\n\n\n
Lesson 9: Deriving the Vertex Form of the Equation of a Parabola<\/a><\/h5>\n\n\n\nSasha and Keoni develop the vertex form of the equation of a parabola as y = (x\u2013h)2<\/sup>\/(4p) + k where the (h,k) is the vertex and p the distance from the vertex to the focus.<\/p>\n\n\n\nLesson 10: Getting and Using Geometric Information<\/a><\/h5>\n\n\n\nGiven 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.<\/p>\n\n\n\n
Sasha and Keoni develop the vertex form of the equation of a parabola as y = (x\u2013h)2<\/sup>\/(4p) + k where the (h,k) is the vertex and p the distance from the vertex to the focus.<\/p>\n\n\n\n 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.<\/p>\n\n\n\nLesson 10: Getting and Using Geometric Information<\/a><\/h5>\n\n\n\n
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