 # C.P. Algebra II The Conic Sections Index The Conics The Conics Translations Completing the Square Completing the Square Classifying Conics Classifying.

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C.P. Algebra II

The Conic Sections Index The Conics The Conics Translations Completing the Square Completing the Square Classifying Conics Classifying Conics

The Conics Parabola Circle Ellipse Hyperbola Click on a Photo Back to Index Back to Index

The Parabola A parabola is formed when a plane intersects a cone and the base of that cone

Parabolas  A Parabola is a set of points equidistant from a fixed point and a fixed line. l The fixed point is called the focus. l The fixed line is called the directrix.

Parabolas Around Us

Parabolas FOCUS Directrix Parabola

Standard form of the equation of a parabola with vertex (0,0) EquationEquation FocusFocus DirectrixDirectrix AxisAxis x 2 =4pyx 2 =4py (0,p)(0,p) y = -py = -p y 2 =4pxy 2 =4px (p,0)(p,0) y = py = p

To Find p 4p is equal to the term in front of x or y. Then solve for p. Example: x 2 =24y 4p=24p=6

Examples for Parabolas Find the Focus and Directrix Example 1 y = 4x 2 x 2 = ( 1 / 4 )y 4p = 1 / 4 p = 1 / 16 FOCUS (0, 1 / 16 ) Directrix Y = - 1 / 16

Examples for Parabolas Find the Focus and Directrix Example 2 x = -3y 2 y 2 = ( -1 / 3 )x 4p = -1 / 3 p = -1 / 12 FOCUS ( -1 / 12, 0) Directrix x = 1 / 12

Examples for Parabolas Find the Focus and Directrix Example 3 (try this one on your own) y = -6x 2 FOCUS???? Directrix????

Examples for Parabolas Find the Focus and Directrix Example 3 y = -6x 2 FOCUS (0, - 1 / 24 ) Directrix y = 1 / 24

Examples for Parabolas Find the Focus and Directrix Example 4 (try this one on your own) x = 8y 2 FOCUS???? Directrix????

Examples for Parabolas Find the Focus and Directrix Example 4 x = 8y 2 FOCUS (2, 0) Directrix x = -2

Parabola Examples Now write an equation in standard form for each of the following four parabolas

Write in Standard Form Example 1 Focus at (-4,0) Identify equation y 2 =4px p = -4 y 2 = 4(-4)x y 2 = -16x

Write in Standard Form Example 2 With directrix y = 6 Identify equation x 2 =4py p = -6 x 2 = 4(-6)y x 2 = -24y

Write in Standard Form Example 3 (Now try this one on your own) With directrix x = -1 y 2 = 4x

Write in Standard Form Example 4 (On your own) Focus at (0,3) x 2 = 12y Back to Conics Back to Conics

Circles A Circle is formed when a plane intersects a cone parallel to the base of the cone.

Circles

Standard Equation of a Circle with Center (0,0)

Circles & Points of Intersection Distance formula used to find the radius

Circles Example 1 Write the equation of the circle with the point (4,5) on the circle and the origin as it’s center.

Example 1 Point (4,5) on the circle and the origin as it’s center.

Example 2 Find the intersection points on the graph of the following two equations

Now what??!!??!!??

Example 2 Find the intersection points on the graph of the following two equations Plug these in for x Plug these in for x.

Example 2 Find the intersection points on the graph of the following two equations Back to Conics Back to Conics

Ellipses

Ellipses Examples of Ellipses Examples of Ellipses

Ellipses Horizontal Major Axis

FOCI (-c,0) & (c,0) CO-VERTICES (0,b)& (0,-b) CENTER (0,0) Vertices (-a,0) & (a,0)

Ellipses Vertical Major Axis

FOCI (0,-c) & (0,c) CO-VERTICES (b, 0)& (-b,0) Vertices (0,-a) & (0, a) CENTER (0,0)

Ellipse Notes l Length of major axis = a (vertex & larger #) l Length of minor axis = b (co-vertex & smaller#) l To Find the foci (c) use: c 2 = a 2 - b 2

Ellipse Examples Find the Foci and Vertices

Write an equation of an ellipse whose vertices are (-5,0) & (5,0) and whose co-vertices are (0,-3) & (0,3). Then find the foci.

Write the equation in standard form and then find the foci and vertices.

Back to the Conics Back to the Conics

The Hyperbola

Hyperbola Examples

Hyperbola Notes Horizontal Transverse Axis Center (0,0) Vertices (a,0) & (-a,0) (-a,0) Foci (c,0) & (-c, 0) (-c, 0) Asymptotes

Hyperbola Notes Horizontal Transverse Axis Equation

To find asymptotes

Hyperbola Notes Vertical Transverse Axis Center (0,0) Vertices (a,0) & (-a,0) (-a,0) Foci (c,0) & (-c, 0) (-c, 0) Asymptotes

Hyperbola Notes Vertical Transverse Axis Equation

To find asymptotes

Write an equation of the hyperbola with foci (-5,0) & (5,0) and vertices (-3,0) & (3,0) a = 3 c = 5

Write an equation of the hyperbola with foci (0,-6) & (0,6) and vertices (0,-4) & (0,4) a = 4 c = 6 The Conics The Conics

Translations Back What happens when the conic is NOT centered on (0,0)? Next

Translations Circle Next

Translations Parabola Next or Horizontal Axis Vertical Axis

Translations Ellipse Next or

Translations Hyperbola Next or

Translations Identify the conic and graph Next r=3 center (1,-2)

Translations Identify the conic and graph Next

Translations Identify the conic and graph Next center asymptotes vertices

Translations Identify the conic and graph center Conic Back to Index Back to Index

Completing the Square Here are the steps for completing the square Steps 1)Group x 2 + x, y 2 +y move constant 2)Take # in front of x, ÷2, square, add to both sides 3)Repeat Step 2 for y if needed 4)Rewrite as perfect square binomial Next

Completing the Square Circle: x 2 +y 2 +10x-6y+18=0 x 2 +10x+____ + y 2 -6y=-18 (x 2 +10x+25) + (y 2 -6y+9)=-18+25+9 (x+5) 2 + (y-3) 2 =16 Center (-5,3)Radius = 4 Next

Completing the Square Ellipse: x 2 +4y 2 +6x-8y+9=0 x 2 +6x+____ + 4y 2 -8y+____=-9 (x 2 +6x+9) + 4(y 2 -2y+1)=-9+9+4 (x+3) 2 + (y-1) 2 =4 C: (-3,1) a=2, b=1 Index

Classifying Conics

Given in General Form Next

Classifying Conics Given in General Form Examples

Classifying Conics Given in general form, classify the conic Ellipse Next

Classifying Conics Given in general form, classify the conic Parabola Next

Classifying Conics Given in general form, classify the conic Hyperbola Next

Classifying Conics Given in general form, classify the conic Hyperbola Back to Index Back to Index

Classifying Conics Given in General Form Then OR If A = C Ellipse Circle Back

Classifying Conics Given in General Form Then Back

Classifying Conics Given in General Form Then Hyperbola Back

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