parabolas and quadratic equations

Armando Martinez-CruzAmartinez-cruz@fullerton.edu

Garrett Delkgdelk71687@csu.fullerton.edu

Department of MathematicsCSU Fullerton

Presented at 2013 CMC Conference

Palm Springs, CA

Parabolas and Quadratic Equations

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

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.

Introduction to Software

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

Constructing • Points • Segments• Lines• Perpendicular Lines

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.

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.

Prove

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

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?

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.

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

Equation of the Parabola Function - I

Distance to Focus = Distance to directrix

. or

.

Equation of the Parabola Function - II

• See Attached Text

Sliders

• Investigation of

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?

Questions