M3CSD6 Have out: Bellwork: red pen, highlighter, GP notebook, calculator, ruler M3CSD6 Have out: Bellwork: Graph the hyperbola. Identify the vertices, foci, and asymptotes. –10 x y 10 +1 +2 hyperbola +1 +1 c2 = a2 + b2 +1 c2 = 9 + 36 Center: (0, 0) (0, 3) c2 = 45 a = 3 +1 (–6,0) (6,0) b = 6 +1 c ≈ 6.71 +1 +1 c = 6.71 +1 foci: (0, ± 6.71) +2 (0,–3) vertices: (0, ±3) +2 +1 +2 box & asymptotes +1 asymptotes: total: +2

Conic Sections: Hyperbolas with Center (h, k) Standard Form of Equation Graph (sketch) Direction of Transverse Axis Center Foci Vertices Length of Transverse Axis Length of Conjugate Axis Equations of Asymptotes horizontal vertical (h, k) (h, k) (h±c, k) (h, k±c) (h±a, k) (h, k±a) 2a 2a 2b 2b

Which variable is positive? x So, it’s a horizontal hyperbola a) I. Graph each hyperbola. Label the vertices and foci. Determine the equations of the asymptotes. Which variable is positive? x So, it’s a horizontal hyperbola a) –10 x y 10 Center: (–1, –3) c2 = a2 + b2 a = 2 c2 = 4 + 9 b = 3 c2 = 13 c = 3.61 foci: (–1±3.61, –3) c ≈ 3.61 (–4.61, –3) and (2.61, –3) (–3,–3) (1, –3) vertices: (–3,–3) and (1,–3) asymptotes:

Which variable is positive? y So, it’s a vertical hyperbola b) I. Graph each hyperbola. Label the vertices and foci. Determine the equations of the asymptotes. Which variable is positive? y So, it’s a vertical hyperbola b) –10 x y 10 Center: (–2, 4) Be careful when you write the center. Look VERY carefully at the equation. c2 = a2 + b2 (–2,8) a = 4 c2 = 16 + 9 b = 3 c2 = 25 c = 5 (–2,0) foci: (–2, 4±5) c = 5 (–2, –1) and (–2, 9) vertices: (–2, 0) and (–2, 8) asymptotes:

Take a few minutes to complete Part I.

center: (5, –5) a = 3 b = 2 c = equation: (x – 5)2 (y + 5)2 32 22 II. Write the standard form equation for each hyperbola. y x –2 12 2 –12 a) The hyperbola opens left and right. Therefore, x is positive. “a” will be under x. a center: (5, –5) b a = 3 b = 2 c = equation: (x – 5)2 (y + 5)2 32 22

center: (6, 3) a = 4 b = 3 c = 5 equation: (y – 3)2 (x – 6)2 42 32 II. Write the standard form equation for each hyperbola. d) x y –1 13 10 –4 The hyperbola opens up and down. Therefore, y is positive. “a” will be under y. a c center: (6, 3) a = 4 b = 3 c = 5 equation: (y – 3)2 (x – 6)2 42 32

Take a few minutes to complete Part II.

III. Write the standard form equation for each hyperbola. Group the x’s and y’s together just like we do for circles and ellipses. Notice the y’s. Recall that subtracting is the same thing as adding a negative. Complete the square. To complete the square for the y’s, first factor out –4. What number is really “added” by the y’s??? 9 –4 – 4y + 4 9 –16 Hyperbolas must equal 1, so divide everything by 4. 4 4 4

III. Write the standard form equation for each hyperbola. Group the x’s and y’s together just like we do for circles and ellipses. Notice the y’s are first since y2 is positive. Note how the x’s are “added.” Complete the square. To complete the square for the x’s, first factor out –3. What number are we really adding to each side? 8 – 4y + 4 –3 + 4x + 4 32 –12 Hyperbolas must equal 1, so divide everything by 24. 24 24 24

Complete today’s packet.