Deriving the Equation of a Parabola From a Focus and Directrix
For a complete lesson plan, please click the button on the right, as well as use the material below.
For a list of activities, descriptions, and teacher guidelines for this lesson, continue to scroll down on this page. |
CCGPS Standards
MCC9-12.G.GPE.2 - Derive the equation of a parabola given a focus and directrix
NCTM Practice Standards
MCC9-12.G.GPE.2 - Derive the equation of a parabola given a focus and directrix
NCTM Practice Standards
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with Mathematics.
- Use appropriate tools strategically.
- Attend to precision.
Activity 1:
Before coming to class the day of this lesson, students should have watched the video below and filled out the "Constructing and Defining a Parabola" worksheet on the right.
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If video does not load below, click the button on the right to take you to the video on Youtube.com
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Activity 2:
Students should work in pairs and use the two applets below to explore the relationship between the parameter "a" and the distance between the focus and the vertex, as well as the shortest distance between the vertex and the focus. Have the students use and complete the "Exploration of Parameter a" document on the right.
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For a link to the GeoGebra applet below, click the button to the right. Also, students should be aware that the applet is graphing the function y-k=a(x-h)^2.
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Teachers should check each pairs work for #10 on the worksheet, evaluating depth of understanding and correctness before allowing the pair to move on to activity 3.
Activity 3:
For this activity, students should work with their partner and use an applet created by Khan Academy to assess how well they understand the different parameters of a quadratic in vertex form. The applet will look like the picture below.
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For this applet, students are asked to drag the focus and directrix around so that the graph corresponds with the equation given (underneath the directions). The equation found in the Answer Box is the equation of the graph currently shown. So students can watch this equation change dynamically as they move the focus and directrix around.
The applet allows students to either be given a hint, or to check their answer. It also tracks whether a students has gotten the answer correct or not, and suggests that the students get 5 correct in a row.
The applet allows students to either be given a hint, or to check their answer. It also tracks whether a students has gotten the answer correct or not, and suggests that the students get 5 correct in a row.
Activity 4:
For this activity, students will work individually and it will serve as a ticket-out-the-door assessment activity. If time does not permit for this, the students can complete the activity at home as long as they have internet access.
Now that the students are familiar with the vertex form of a parabola, understand the relationship between the parameter a and the distance between the focus and the vertex (and the distance between the vertex and directrix), and how the equation of a parabola defines the shape of the curve, they should be able to derive the equation of a parabola given only the focus and directrix.
Now that the students are familiar with the vertex form of a parabola, understand the relationship between the parameter a and the distance between the focus and the vertex (and the distance between the vertex and directrix), and how the equation of a parabola defines the shape of the curve, they should be able to derive the equation of a parabola given only the focus and directrix.
To test this skill, students will use another applet from Khan Academy that looks like the picture below.
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This applet is a little harder to work with, because students have to estimate where they believe the focus and directrix should be. In order for students to understand how to work this applet, refer them to a video showing them how to use the applet by clicking here.
Once the student discovers the correct location of the focus and directrix, they should write the equation of the parabola in the answer box.
Again, Khan Academy keeps track of students correct and incorrect answers, and suggests that they get 3 correct answers in a row. In order to show the teacher that the student really did get three correct answers in a row, The student should take a screenshot of their screen so that the teacher can see the three check marks. Then the student should email this photo in, or use another technology such as Evernote or Padlet (if the instructor has established these within their classroom).
Once the student discovers the correct location of the focus and directrix, they should write the equation of the parabola in the answer box.
Again, Khan Academy keeps track of students correct and incorrect answers, and suggests that they get 3 correct answers in a row. In order to show the teacher that the student really did get three correct answers in a row, The student should take a screenshot of their screen so that the teacher can see the three check marks. Then the student should email this photo in, or use another technology such as Evernote or Padlet (if the instructor has established these within their classroom).