parabolas and quadratic equations
DESCRIPTION
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 PresentationTRANSCRIPT
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Armando [email protected]
Garrett [email protected]
Department of MathematicsCSU Fullerton
Presented at 2013 CMC Conference
Palm Springs, CA
Parabolas and Quadratic Equations
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Agenda• Welcome• CCSS• Intro to Software • Parabolas - Locus• Sliders• Questions
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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.
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Introduction to Software
• Points• Segments• Midpoint• Perpendicular Lines• Locus• Sliders
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Constructing • Points • Segments• Lines• Perpendicular Lines
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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.
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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.
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Prove
• Point F is equidistant to the directrix and the focus.
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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?
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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.
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Since the distance to any point A on the circle to the Center is r…
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Equation of the Parabola Function - I
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Distance to Focus = Distance to directrix
. or
.
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Equation of the Parabola Function - II
• See Attached Text
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Sliders
• Investigation of
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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?
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Questions