5.04b - Hyperbola Transformed

University:
High School
Course:
High School Physics
Academic year:

2024
Views:

170

Pages:

15
Author:

1 1

5.04b Hyperbola Transformed Transformed Hyperbola • Foci on Horizontal (𝑥 − ℎ)2 𝑦−𝑘 2 − =1 2 2 𝑎 𝑏 • Foci on Vertical Axis (𝑥 − ℎ)2 𝑦−𝑘 2 − = −1 2 2 𝑎 𝑏 Asymptotes: 𝑏 1. Slope: ± 𝑎 2. Initial Value: use (ℎ, 𝑘) as (𝑥, 𝑦) in 𝑏 = 𝑦 − 𝑎𝑥 Transformed Inequalities Horizontal (𝑥 − ℎ)2 𝑦−𝑘 − 2 𝑎 𝑏2 ≤ ≥ 2 ໆ1 Vertical (𝑥 − ℎ)2 𝑦−𝑘 − 2 𝑎 𝑏2 2 INSIDE OUTSIDE OUTSIDE INSIDE ໆ −1 Other important points: 𝐴′ ℎ − 𝑎, 𝑘 𝑎𝑛𝑑 𝐴(ℎ + 𝑎, 𝑘) 𝐹′ ℎ − 𝑐, 𝑘 𝑎𝑛𝑑 𝐹 ℎ + 𝑐, 𝑘 𝐶𝑒𝑛𝑡𝑒𝑟 (ℎ, 𝑘) 𝑐 2 = 𝑎2 + 𝑏2 𝐵′ ℎ, 𝑘 − 𝑏 𝑎𝑛𝑑 𝐵(ℎ, 𝑘 + 𝑏) 𝐹′ ℎ, 𝑘 − 𝑐 𝑎𝑛𝑑 𝐹 ℎ, 𝑘 + 𝑐 𝒃 𝒚𝟏 = 𝒙 + 𝑰𝑽𝟏 𝒂 Graphing OR Finding Rule you need to understand the parts of the hyperbola 𝒃 𝒚𝟐 = − 𝒙 + 𝑰𝑽𝟐 𝒂 You need to understand the parts of the hyperbola 𝒃 𝒚𝟏 = 𝒙 + 𝑰𝑽𝟏 𝒂 𝒃 𝒚𝟐 = − 𝒙 + 𝑰𝑽𝟐 𝒂 REMEMBER Horizontal Vertical • • • • • • • • • • 𝐶 ℎ, 𝑘 𝐴(ℎ + 𝑎, 𝑘) 𝐴′ ℎ − 𝑎, 𝑘 𝐹 ℎ + 𝑐, 𝑘 𝐹′(ℎ − 𝑐, 𝑘) 2 𝐶 ℎ, 𝑘 𝐵(ℎ, 𝑘 + 𝑏) 𝐵′ ℎ, 𝑘 − 𝑏 𝐹 ℎ, 𝑘 + 𝑐 𝐹′(ℎ, 𝑘 − 𝑐) 2 𝑤ℎ𝑒𝑟𝑒 𝑐 = 𝑎 + 𝑏 2 To Graph 1.Plot Center 2.Create box using 𝑎 and 𝑏 3.Draw asymptotes 4.Draw Hyperbola (horizontal/vertical based on rule) 5.Identify vertices and focal points Find the Rules of Asymptotes 1. Find slope 𝑏 ± 𝑎 2. Use (ℎ, 𝑘) as a point on the line to solve for initial value 3. ASYMPTOTES: 𝑏𝑥 𝑦1 = + 𝑖𝑣1 𝑎 𝑏𝑥 𝑦2 = − + 𝑖𝑣2 𝑎 Example: 𝑥+2 2 16 − 𝑦−1 2 4 = −1 𝑎 = 4, 𝑏 = 2 • Slope: 1 ± 2 • Use ℎ, 𝑘 : −2,1 𝑥 𝑦1 = + 𝐼𝑉1 2 −2 1=− + 𝐼𝑉2 2 0 = 𝐼𝑉2 2 1 = − + 𝐼𝑉1 2 2 = 𝐼𝑉1 𝑥 𝑦2 = − + 𝐼𝑉2 2 𝑥 𝑦1 = + 2 2 𝑥 𝑦2 = − 2 Find Intersections of Hyperbola 1. Sub in value 2. Solve the resulting equation – It will always be a quadratic EX 𝑥 = 0: 𝑥+2 16 2 𝑦−1 − 4 2 = −1 Example: 𝑥+2 2 16 − 𝑦−1 2 4 = −1 0+2 2 𝑦−1 2 − = −1 16 4 4 𝑦−1 2 − = −1 16 4 4 − 4 𝑦 − 1 2 = −16 Use the QUAD FORM 4 − 4 𝑦 2 − 2𝑦 + 1 = −16 𝑦1 = 1 − 5 ≈ −1.235 𝑦2 = 1 + 5 ≈ 3.236 −4 𝑦 2 − 2𝑦 + 1 = −20 𝑦 2 − 2𝑦 + 1 = 5 𝑦 2 − 2𝑦 − 4 = 0 −𝑏 ± 𝑏 2 − 4𝑎𝑐 𝑦= 2𝑎 General Form to Standard Form • COMPLETE THE SQUARE 4𝑥 2 − 25𝑦 2 − 24𝑥 − 200𝑦 − 264 = 0 4𝑥 2 − 24𝑥 − 25𝑦 2 − 200𝑦 = 264 4(𝑥 2 − 6𝑥) − 25(𝑦 2 + 8𝑦) = 264 4 𝑥 2 − 6𝑥 + 9 − 25 𝑦 2 + 8𝑦 + 16 = 264 + 36 − 400 4 𝑥 − 3 2 − 25 𝑦 + 4 2 = −100 4 𝑥 − 3 2 25 𝑦 + 4 2 − = −1 100 100 𝑥−3 2 𝑦+4 2 − = −1 25 4 Homework • Page 501 #21d, 22ace Homework • Page 501 #24a