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Armando Martinez-Cruz [email protected] Garrett Delk [email protected] Department of Mathematics CSU Fullerton Presented at 2013 CMC Conference Palm Springs, CA Parabolas and Quadratic Equations

Parabolas and Quadratic Equations

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Parabolas and Quadratic Equations. Armando Martinez-Cruz [email protected] Garrett Delk [email protected] Department of Mathematics CSU Fullerton Presented at 2013 CMC Conference Palm Springs, CA. Agenda. Welcome CCSS Intro to Software Parabolas - Locus - PowerPoint PPT Presentation

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Page 1: Parabolas and Quadratic Equations

Armando [email protected]

Garrett [email protected]

Department of MathematicsCSU Fullerton

Presented at 2013 CMC Conference

Palm Springs, CA

Parabolas and Quadratic Equations

Page 2: Parabolas and Quadratic Equations

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

Page 3: Parabolas and Quadratic Equations

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.

Page 4: Parabolas and Quadratic Equations

Introduction to Software

• Points• Segments• Midpoint• Perpendicular Lines• Locus• Sliders

Page 5: Parabolas and Quadratic Equations
Page 6: Parabolas and Quadratic Equations

Constructing • Points • Segments• Lines• Perpendicular Lines

Page 7: Parabolas and Quadratic Equations

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.

Page 8: Parabolas and Quadratic Equations

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.

Page 9: Parabolas and Quadratic Equations
Page 10: Parabolas and Quadratic Equations
Page 11: Parabolas and Quadratic Equations

Prove

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

Page 12: Parabolas and Quadratic Equations

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?

Page 13: Parabolas and Quadratic Equations

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.

Page 14: Parabolas and Quadratic Equations

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

Page 15: Parabolas and Quadratic Equations

Equation of the Parabola Function - I

Page 16: Parabolas and Quadratic Equations

Distance to Focus = Distance to directrix

. or

.

Page 17: Parabolas and Quadratic Equations

Equation of the Parabola Function - II

• See Attached Text

Page 18: Parabolas and Quadratic Equations

Sliders

• Investigation of

Page 19: Parabolas and Quadratic Equations

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?

Page 20: Parabolas and Quadratic Equations

Questions