The Wonderful World of Conics Parabola Circle Ellipse Hyperbola By: Ethan Ottone, Ace Holland, and Jeremy Heaggans

What is a Conic?  The definition of a conic is a curve obtained as the intersection of a cone with a plane.  There are four different types of conics: parabolas, circles, ellipses, and hyperbolas.

Parabolas  A parabola represents the locus of points in a plane that are equidistant from a fixed point, which is called the focus, and a specific line, called the directrix.

Standard Form of Equations for Parabolas Standard Form of Equations for Parabolas click on hyperlink for parabolas video!  Orientation: opens vertically  Vertex: (h,k)  Focus: (h, k+p)  Axis of Symmetry a: x=h  Direc trix d: y=k-p  If the P value is positive then it opens up but if its negative it opens down  Equation: (x-h)^2= 4p(y-k)  Orientation: opens horizontally  Vertex: (h,k)  Focus: (h+p,k)  Axis of Symmetry a: y=k  Directrix d: x=h-p  If the P value is negative then it opens left and if its positive it opens right  Equation: (y-h)^2= 4p(x-k)

Example 1  For (y+5)^2= -12(x- 2), identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.

Example 2 How tall is one of natures many parabolas the Rainbow? (maximum of the parabola) Answer: maximum= 40

Click on Hyperlink for Real world example!!  Sure you may think that this PowerPoint is boring because of these dull backgrounds…. But guess what!!?? Parabolas can be used in the REAL WORLD!!!!! AWSOME!!!! Sure you may think that this PowerPoint is boring because of these dull backgrounds…. But guess what!!?? Parabolas can be used in the REAL WORLD!!!!! AWSOME!!!!

Circles  Circles are formed when a plane intersects a cone parallel to the base of the cone.

Standard Equation of a Circle with center (0,0) Standard Equation of a Circle with center (0,0) click on hyperlink above for circles video!  X^2 + Y^2 = R^2

Example 1

Real World Examples  Truth is circles are EVERYWHERE!!! So of course you can use them in the real world!!! Examples of circles in real life:  Clocks  Tires  Plates  DVD’s  Etc…

Ellipses  Formed when a plane cuts the axis of a cone and the surface of a cone.

The Equations of an Ellipse The Equations of an Ellipse click on hyperlink above for ellipse video!  Horizontal Major Axis  X^2/a^2 + y^2/b^2=1  Foci: (-c, 0) &(c, 0)  Co- Vertices: (0, b) & (0,-b)  Vertices: (-a,0) & (a, 0)  Vertical Major Axis  X^2/b^2 + y^2/a^2= 1  Foci: (0, -c) & (o,c)  Co- Vertices: (b,0) & (-b, 0)  Vertices: (0, -a) & (0,a)

Example 1  Graph the ellipse equation: x^2/25 + Y^2/16 = 1

Click on Hyperlink for Real world example!!  “Ellipses look like ovals we can never use them in the real world!!” Which is what you are probably saying to yourself right now. But real talk my fellow student, you can! “Ellipses look like ovals we can never use them in the real world!!” Which is what you are probably saying to yourself right now. But real talk my fellow student, you can!

Hyberbolas  Is formed by the intersection of the plane with a cone when the plane makes a greater angle with the base than does the generator of the cone.

The Equations of a Hyperbola The Equations of a Hyperbola click on Hyperlink for hyperbola video!  Horizontal Transverse Axis  (x-h)^2 /a^2 – (y-k)^2/b^2=1  Foci: ( h +/- c, k)  Vertices: ( h +/- a, k)  Center: (h,k)  Asymptotes: y-k= +/- b/a(x-h)  Abc Relationship: c^2 = a^2+b^2 or c^2 = square root of a^2+b^2  Vertical Transverse Axis  (y-k)^2 /a^2 – (x-h)^2 / b^2 =1  Foci : (h +/- k, c)  Vertices : (h +/- k, a)  Center: (h,k)  Asymptotes: y-k= +/- a/b(x-h)  Abc Relationship: c^2 = a^2+b^2 or c^2 = square root of a^2+b^2

Example 1  Graph the hyperbola equation: (x-2)^2 /9 – (y-1)^2 /16 = 1

Click on Hyperlink for Real world example!!  Who needs hyperbolas!!!??? You might being saying this in your head right now but hyperbolas can be found and used in the REAL WORLD!!! Who needs hyperbolas!!!??? You might being saying this in your head right now but hyperbolas can be found and used in the REAL WORLD!!!