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) 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 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. 15.