Modeling with Conic Sections and Polar Equations Module Overview

Introduction

As you continue your study of Precalculus, we will begin our exploration of conic sections. You've probably studied circles and parabolas before, but ellipses and hyperbolas will be new! A conic section is formed by the intersection of a plane and a cone as shown in the image!

Next, we'll revisit complex numbers and discover how they are applied real world applications. In the field of Physics, complex numbers are used in electronic and electromagnetism engineering and quantum mechanics. Mathematicians use complex numbers to solve differential equations and apply it in many analysis fields. In this module, we will begin with a review of operations with complex numbers, then move on to study the graphical representation of complex numbers in both rectangular and polar form.

Essential Questions

How do I graph circles, parabolas, ellipses, and hyperbolas?
How are ellipses and hyperbolas defined in relation to their foci?
How can conic sections be graphed when given in general form?
How do I write the equation of a conic section given two features?
How can I represent complex numbers graphically?
How does the complex plane show addition, subtraction, and conjugation of complex numbers?
What are two ways to represent complex numbers?
When given two points on the complex plane, what does it mean to find the distance between them and the midpoint of the segment connecting them?

Key Terms

The following key terms will help you understand the content in this module.

Cone - a three-dimensional figure with a circular or elliptical base and one vertex

Coplanar - set of points, lines, rays, line segments, etc. that lie in the same plane

Ellipse - a curved line forming a closed loop. It is the locus of points for which the sum of the distances from two fixed points (foci) to every point on the curve is constant.

Focus - one of the fixed points from which the distances to any point of a given curve

Hyperbola - a plane curve having two branches, formed by the intersection of a plane with both halves of a right circular cone at an angle parallel to the axis of the cone. It is the locus of points for which the difference of the distances from the two given points is a constant.

Locus of Points - a group of points that share a property

Complex Numbers - A class of numbers including purely real numbers a, purely imaginary number bi, & numbers with both real and imaginary parts a + bi

Rectangular form of a complex number - a + bi

cisθ - Shorthand for equation image indicator $LaTeX: \cos\theta+i\sin\theta$

Polar Form of a Complex Number - equation image indicator $LaTeX: r\left(\cos\theta+i\sin\theta\right)=r\left(cis\theta\right)$

Complex Conjugate of z = a + bi - $LaTeX: \overline{z}=a+bi$

Modulus of a Complex Number - the distance between a number and 0 when plotted on the complex plane. equation image indicator $LaTeX: \left|z\right|=\left|a+bi\right|=\sqrt[]{a^2+b^2}$ . The modulus is also referred to as the absolute value.

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