About This Chapter
Standard: Derive the equation of a parabola given a focus and directrix. (CCSS.Math.Content.HSG-GPE.A.2)
Standard: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. (CCSS.Math.Content.HSG-GPE.A.3)
About This Chapter
This standard focuses on deriving the equations of various conic sections, including parabolas, ellipses and hyperbolas. In addition to deriving equations, students who understand the standard will be able to find the focus, directrix and definition of various conic shapes.
The lessons in this standard show students how to:
- Find the focus and directrix of a parabola
- Uncover the focus and definition of an ellipse and a hyperbola
- Discover the equation of a parabola from the focus and directrix
- Derive the equation of an ellipse or a hyperbola from the foci
Individuals with mastery of this standard will know how to find the focus and directrix of a parabola, as well as the focus and definition of an ellipse and a hyperbola. They will also be able to find the equation of a parabola, ellipse and hyperbola from the focus.
How to Use These Lessons in Your Classroom
Below are tips for utilizing these lessons in the classroom to complement instruction in the CCSS.Math.Content.HSG-GPE.A.2 and CCSS.Math.Content.HSG-GPE.A.3 standards:
Working with Parabolas Lessons
Assign the chapters The Focus and Directrix of a Parabola and Finding the Equation of a Parabola from the Focus and Directrix for homework. Have students break into small groups and switch off between finding the focus and directrix from a parabola equation and its opposite operation of using the parabola equation to discover the focus and directrix.
Using Ellipses and Hyperbolas
Split students into two groups: one that covers ellipses and another group that covers hyperbolas. Have each group answer questions requiring them to derive the equation and find the focus and definition of their respective conic section. After they've answered a few questions, tell the students to switch.
Conic Section Practice Lessons
Have students watch the Practice with Conic Sections video lesson and answer all example questions. Go over these practice quiz answers in class.
1. The Focus and Directrix of a Parabola
A parabola is the curve of a line on the graph representing the quadratic equation, shaped like a 'U'. Learn how the focus and the directrix help define the parabola through examples with visual representations.
2. Finding the Equation of a Parabola from the Focus and Directrix
The equation for a parabola curve can be found from the focus and directrix. Discover how to find the standard form of the parabola equation from the focus and directrix, as well as two other forms of the equation: intercept form and vertex form.
3. Foci and the Definitions of Ellipses and Hyperbolas
Foci are points located on the curves of ellipses and hyperbolas. Study the definition of ellipses and hyperbolas, learn the foci of these shapes, and have a look at examples for further understanding.
4. Derive the Equation of an Ellipse from the Foci
An ellipse refers to all points in a set where the sum of distances from the foci or two fixed points is constant. Study the definition of an ellipse and foci, and learn how to derive an equation from the horizontal and vertical major axis.
5. Derive the Equation of a Hyperbola from the Foci
A hyperbola can be shaped on a graph similar to a butterfly's wing and has a formula derived from the foci, which are two points inside the branches at a fixed distance from the center, and other points. Learn how to derive the equation of a hyperbola from just the foci through practice examples.
6. Practice with the Conic Sections
Conic sections are shapes made cutting a 3D cone at specific angles. Get some practice with the conic sections by looking at examples which include circles, ellipses, parabolas, and hyperbolas.
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