Math III TITD Midpoint Distance Completing the Square

Math III

TITD Midpoint Distance Completing the Square

Conic Sections The intersection of a plane and a cone. http: //www. math. odu. edu/cbii/calcanim/consec. avi

STANDARDS MGSE 9 -12. A. REI. 7 v Solve a simple system consisting of a linear equation and quadratic equation in two variables algebraically and graphically. v

STANDARDS MGSE 9 -12. G. GPE. 2 v Derive the equation of a parabola given a focus and directrix. v MGSE 9 -12. G. GPE. 3 v Derive the equation of ellipses and hyperbolas given to foci for the ellipse, and two directrices for the hyperbolas v

ESSENTIAL QUESTIONS 1) How do I indentify the characteristics of circles from equations? 2) What characteristics of circles are necessary to graph and write the equations of circles?

KEY VOCABULARY Ø Ø Ø Ø Ø Cone Coplanar Focus Directrix Circle Equidistance Center Radius General form Standard form

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

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

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

Parabolas Around Us Back to Conics

§ Parabolas A Parabola is a set of points equidistant from a fixed point and a fixed line. • The fixed point is called the focus. • The fixed line is called the directrix. Back to Conics

Parabolas Parabola FOCUS Directrix Back to Conics

Standard form of the equation of a parabola with vertex (0, 0) • Equation • Focus • Directrix • x 2=4 py • (0, p) • y = -p • y 2=4 px • (p, 0) • x = p Back to Conics • Axis

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

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

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

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

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

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

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

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

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

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

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

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

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

Circles in real life Back to Conics

Standard Equation of a Circle with Center (0, 0) Back to Conics

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

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

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

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

Now what? ? !!? ? Back to Conics

Example 2 Find the intersection points on the graph of the following two equations Back to Conics Substitute these in for x.

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

Ellipses An ellipses is formed when a plane intersects a cone without being parallel or perpendicular to the base of the cone. Back to Conics Back to Index

Ellipses Examples of Ellipses Back to Conics

Ellipses Horizontal Major Axis Back to Conics

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

Ellipses Vertical Major Axis Back to Conics

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

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

Ellipse Examples Find the Foci and Vertices Back to Conics

Ellipse Examples Find the Foci and Vertices Back to Conics

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. Back to Conics

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

The Hyperbola An hyperbola is formed when a plane intersects a cone parallel to the axis of the cone. Back to Conics Back to Index

Hyperbola Examples Back to Conics

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

Hyperbola Notes Horizontal Transverse Axis Equation Back to Conics

Hyperbola Notes Horizontal Transverse Axis To find asymptotes Back to Conics

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

Hyperbola Notes Vertical Transverse Axis Equation Back to Conics

Hyperbola Notes Vertical Transverse Axis To find asymptotes Back to Conics

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

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

Back to the Conics

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

Translations Circle Back to Index Next

Translations Parabola Horizontal Axis or Vertical Axis Back to Index Next

Translations Ellipse or Back to Index Next

Translations Hyperbola or Back to Index Next

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

Translations Identify the conic and graph Back to Index Next

Translations Identify the conic and graph center asymptotes Back vertices Back to Index Next

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

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 Back to Index Next

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

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

Classifying Conics

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

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

Classifying Conics Given in General Form Then Back

Classifying Conics Given in General Form Then Hyperbola Back