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Parabolas and Quadratic Equations

Parabolas and Quadratic Equations. Armando Martinez-Cruz Amartinez-cruz@fullerton.edu Garrett Delk gdelk71687@csu.fullerton.edu Department of Mathematics CSU Fullerton Presented at 2013 CMC Conference Palm Springs, CA. Agenda. Welcome CCSS Intro to Software Parabolas - Locus

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Parabolas and Quadratic Equations

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  1. Parabolas and Quadratic Equations Armando Martinez-Cruz Amartinez-cruz@fullerton.edu Garrett Delk gdelk71687@csu.fullerton.edu Department of Mathematics CSU Fullerton Presented at 2013 CMC Conference Palm Springs, CA

  2. Agenda • Welcome • CCSS • Intro to Software • Parabolas - Locus • Sliders • Questions

  3. Parabolas and CCSS • Mathematics » High School: Geometry » Expressing Geometric Properties with Equations • Translate between the geometric description and the equation for a conic section • CCSS.Math.Content.HSG-GPE.A.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. • CCSS.Math.Content.HSG-GPE.A.2 Derive the equation of a parabola given a focus and directrix.

  4. Introduction to Software • Points • Segments • Midpoint • Perpendicular Lines • Locus • Sliders

  5. Constructing • Points • Segments • Lines • Perpendicular Lines

  6. Parabolas as a Locus • The parabola is the locus of all points (x, y) that are equidistant to a fixed line called the directrix, and a fixed point called the focus.

  7. Steps to Construct the Parabola-Locus • Construct a point, A. This is the focus. • Construct line BC (not through A). This is the directrix. • Construct point D (different from A and B) on the directrix. • Construct the perpendicular line to the directrix through D. • Construct segment AD. • Construct the midpoint, E, of segment AD. • Construct the perpendicular bisector of segment AD. • Construct the point of intersection, F, of this perpendicular bisector with the perpendicular to the directrix. • Construct the locus of F when D moves along the directrix.

  8. Prove • Point F is equidistant to the directrix and the focus.

  9. Investigation • Drag the vertex. What happens to the parabola as the vertex move? • Drag the directrix. What happens to the parabola as the directrix move?

  10. The Equation of a Circle A circle is defined as the set of all points (x, y) that are equidistant from a fixed point, (h, k), called the center. The fixed distance is called the radius.

  11. Since the distance to any point A on the circle to the Center is r…

  12. Equation of the Parabola Function - I

  13. . or Distance to Focus = Distance to directrix .

  14. Equation of the Parabola Function - II • See Attached Text

  15. Sliders • Investigation of

  16. An Investigation with the Vertex • The vertex is located at (-b/2a, f(-b/2a)) • Enter d = -b/2a in INPUT box and plot V = (d, f(d)). What happens to the vertex as b moves and a and c remain fix?

  17. Questions

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