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Guided Notes for Unit 4-4: Hyperbola Learning Target: I will graph hyperbolas.

HYPERBOLA TERMS Vertex EQUATION FORM CENTER The “Butterfly” VERTICES CO-VERTICES ASYMPTOTES (h, k ) (h ± a , k) Co-vertex (h, k ± b ) Vertex b a Vertex C=(h , k) Co-vertex

HYPERBOLA TERMS Vertex EQUATION FORM CENTER The “Hourglass” VERTICES CO-VERTICES ASYMPTOTES (h, k ) Vertex (h, k ± a ) (h ± b, k) C=(h , k) Co-vertex a b Co-vertex Vertex

Example 1 The equation of the form 𝑥−ℎ 2 𝑎 2 − 𝑦−𝑘 2 𝑏 2 =1, so Graph 4x2 – 16y2 = 64. 4x2 – 16y2 = 64 – = 1 Rewrite the equation in standard form. x 2 16 y 2 4 Since a2 = 16 and b2 = 4, a = 4 and b = 2. The equation of the form 𝑥−ℎ 2 𝑎 2 − 𝑦−𝑘 2 𝑏 2 =1, so Step 1: Graph the vertices. Since the transverse axis is horizontal, the vertices lie on the x-axis. The coordinates are (±a, 0), or (±4, 0). Step 2: Use the values a and b to draw the central “invisible” rectangle. The lengths of its sides are 2a and 2b, or 8 and 4.

Example 1 Graph 4x2 – 16y2 = 64. Step 3: Draw the asymptotes. The equations of the asymptotes are 𝑦=𝑘 ± 𝑏 𝑎 𝑥−ℎ , which comes out to be 𝑦 = 1 2 𝑥 and 𝑦 = − 1 2 𝑥 the asymptotes contain the diagonals of the central rectangle. Step 4: Sketch the branches of the hyperbola through the vertices so they approach the asymptotes.

Graph and Label b) Find coordinates of vertices, covertices a) GRAPH Center = (-5,-2) Butterfly shape since the x terms come first Since a = 2 and b = 3 Vertices are 2 points left and right from center  (-5 ± 2, -2) CoVertices are 3 points up and down  (-5, -2 ± 3) a) GRAPH Plot Center (-5,-2) a = 2 (go left and right) b = 3 (go up and down)

Graph and Label b) Find coordinates of vertices, covertices, a) GRAPH Center = (-1,3) Hourglass shape since the y terms come first Since a = 2 and b = 4 Vertices are 2 points up and down from center  (-1, 3 ± 2) Covertices are 3 points left and right  (-1 ± 4, 3) a) GRAPH Plot Center (-1,3) a = 2 (go up and down) b = 4 (go left and right)

Assignment Pg. 629 #16-19