UNIT 8 CONIC SECTIONS DAY 3: HYPERBOLA

1 WARM UP  Which equations are ellipses??

1) +9 + 90 +189= 0 2) + − 6 +8=0 3) 4 − 25 + 50 −125= 0 1) Ellipse 4) + + 4 − 2 −4=0 2) Circle 5) −2 − 20 + −46=0 4) Circle 6) 4 + +25−15=0 6) Ellipse 7) 49 +4 −196= 0 7) Ellipse 8) + − 4 − 6 +4=0 8) Circle 9) − 12 + +36=0 10) 64 − 36 + 192 + 36 −441= 0

 Sketch the 4 conic sections and describe how to form them by cutting a

double-napped cone. 2 APPLICATION OF HYPERBOLAS

 Applications of hyperbolas, nuclear power plant cooling towers  https://www.youtube.com/watch?v=EozoaOfYNOI  https://www.youtube.com/watch?v=bzrvBYXURgY

3 HYPERBOLA VOCABULARY

 center: the point (h, k) at the center of a circle, an ellipse, or an hyperbola  transverse axis: line connecting the vertices  conjugate axis: line connecting the co-vertices  vertex: points where the hyperbola makes it’s sharpest turns  co-vertex: an end point of the conjugate axis  focus: a point from which distances are measured in forming a conic; a point at which these distance-lines converge, or "focus” 4  asymptotes: ensure the boundaries of the hyperbola graph HYPERBOLA

 A hyperbola is the locus of points in a plane such that the absolute value of the difference of the distances from two distinct fixed points (foci) is constant

. . -a and a are vertices  -b and b are co-vertices  -c and c are foci  Asymptotes intersect at the vertex 5 HYPERBOLA EQUATIONS

 Standard form: transverse axis (ta) is horizontal transverse axis is vertical     6 GRAPHING HYBERBOLAS

 Graph  Center (-1, 2)    Vertices: (-1,6), (-1,-2)  Co-Vertices: (-4,2), (2,2)  Foci: (-1,7), (-1,-3) 7  Asymptotes= YOU TRY!

 Graph  Center (2, 2)  .   Vertices: (0,2), (4,2)  Co-Verticies: (2,5), (2, -1)  Foci: (5.6,2), (-1,6,2)  Asymptotes: 8 WRITING EQUATIONS FROM A GRAPH

 Write the equation of the hyperbola shown in the graph  From the graph  a= 2, b=3  The center is at (0,3)

  9 YOU TRY!

 Write the equation of the hyperbola shown in the graph  From the graph  a= 4, b=2  The center is at (0, 0)

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10 WRITING EQUATIONS GIVEN CHARACTERISTICS

 Write the equation of the hyperbola with vertices at (0,2) and (4,2) and foci at (-1,2) and (5,2).  Vertices are on the transverse axis therefore the ta is parallel to the x-axis x −x  Using the coordinates of the vertices: a= 2 1  Center is at (2,2) x −x  Using the coordinates of the foci: c= 2 1 

 11 YOU TRY!

 Write the equation of the hyperbola with foci at (2,-5) and (2,3) and vertices at (2,-4) and (2,2).  a= 3, c= 4  Center is at (2,-1)  b2= 16-9=7

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12 CONVERTING BETWEEN FORMS

Find the standard form of the hyperbola from the general form   (Group terms)  (Factor)  (Complete the square)  2

13  COMPARING EQUATIONS

Conic Section Standard Form General Form

Hyperbola** Ellipse Circle

**Notice the negative sign between the squared variables 14 EXIT TICKET

 WebAssign due on Monday, January 22.

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