DERIVING THE GENERAL EQUATION OF A PARABOLA

Transcription

1 DERIVING THE GENERAL EQUATION OF A PARABOLA Standard Addressed in this Task MCC9 12.G.GPE.2 Derive the equation of a parabola given a focus and directrix. Standards for Mathematical Practice 2. Reason abstractly and quantitatively by requiring students to make sense of quantities and their relationships to one another in problem situations. 3. Construct viable arguments and critique the reasoning of others by engaging students on discussion of why they agree or disagree with responses, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 7. Look for and make use of structure by expecting students to apply rules, look for patterns and analyze structure. 8. Look for and express regularity in repeated reasoning by expecting students to understand broader applications and look for structure and general methods in similar situations. Parabolas were studied in the previous unit as quadratic functions where the equations were based on the position of the vertex and additional points were found using values of x on either 2 side of the axis of symmetry. Equations were in the vertex form, y= ax ( h) + k, or in 2 general quadratic form y = ax + bx + c. We will use these forms and expand our study by including the geometric definition of the parabola. Definition: A parabola is the set of all points that are the same distance from a fixed point, the focus, and a fixed line, the directix. Part 1: Finding a specific equation. Consider the parabola below. Notice the vertex at the origin and a point on the parabola (x, y). July 2014 Page 56 of 88

2 Follow the steps to find the equation of the parabola. 1. Write equations for the distance from the focus to (x, y) and the distance from the directrix to (x, y). 2. Because the definition of a parabola says that the distances you wrote in #1 are the same, write an equation stating this fact. 3. Now square both sides to eliminate the radicals. Square the binomials and write your answer below. July 2014 Page 57 of 88

3 4. Collect like terms and solve the equation for y. 5. Write your equation in terms of y below. July 2014 Page 58 of 88

4 Part 2: Writing the general equation for a Parabola. Consider the graph of the parabola below. It s vertex is at the origin, the focus is at (0,p), and the directrix is the line y=-p. Follow the steps to find the equation of the parabola. 1. Write equations for the distance from the focus to (x, y) and the distance from the directrix to (x, y). 2. Because the definition of a parabola says that the distances you wrote in #1 are the same, write an equation stating this fact. July 2014 Page 59 of 88

5 3. Now square both sides to eliminate the radicals. Square the binomials and write your answer below. 4. Collect like terms and solve the equation for y. 5. Write your equation in terms of y below. 6. What does the p represent in your equation? Consider that the vertex of a parabola is not always at the origin. It could be translated to any point on the graph. Use what you know about transformations and your formula above to write the following equations. 7. Vertex at (2, 3); p=4. 8. Vertex at (-4, 8); p= Vertex at (5, -9); p= Vertex at (h, k); p=p. July 2014 Page 60 of 88

6 PARABOLAS IN OTHER DIRECTIONS Standard Addressed in this Task MCC9 12.G.GPE.2 Derive the equation of a parabola given a focus and directrix. Standards for Mathematical Practice 2. Reason abstractly and quantitatively by requiring students to make sense of quantities and their relationships to one another in problem situations. 3. Construct viable arguments and critique the reasoning of others by engaging students on discussion of why they agree or disagree with responses, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 7. Look for and make use of structure by expecting students to apply rules, look for patterns and analyze structure. 8. Look for and express regularity in repeated reasoning by expecting students to understand broader applications and look for structure and general methods in similar situations. Common Student Misconceptions 1. Because new vocabulary is being introduced in this cluster, remembering the names of the conic sections can be problematic for some students. 2. The Euclidean distance formula involves squared, subscripted variables whose differences are added. The notation and multiplicity of steps can be a serious stumbling block for some students. 3. The method of completing the square is a multi-step process that takes time to assimilate. 4. A geometric demonstration of completing the square can be helpful in promoting conceptual understanding. Teacher Notes: This task introduces students to horizontal parabolas. Students should be able to identify horizontal or vertical parabolas from equations without transforming them to standard form. When we studied quadratic functions in the last unit, we used only parabolas that opened up or down. When we use a geometric definition of parabolas, we also can write equations for parabolas that open right or left. 1 2 Consider the parabola y = x. When we compare it to our general form equation, we see that: 4 July 2014 Page 61 of 88