5.05b - Parabola Transformed

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

2024
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144

Pages:

12
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5.05B TRANSFORMED PARABOLA 𝒙=𝒉−𝒄 Horizontal - RIGHT 𝑉(ℎ, 𝑘) 𝐅𝐨𝐜𝐚𝐥 𝐀𝐱𝐢𝐬 𝑑(ℎ − 𝑐, 𝑘) 𝐹(ℎ + 𝑐, 𝑘) 𝒚=𝒌 2 (𝑦 − 𝑘) = 4𝑐 𝑥 − ℎ 𝒙=𝒉+𝒄 Horizontal - LEFT 𝑉(ℎ, 𝑘) 𝐅𝐨𝐜𝐚𝐥 𝐀𝐱𝐢𝐬 𝐹(ℎ − 𝑐, 𝑘) 𝒚=𝒌 (𝑦 − 𝑘)2 = −4𝑐 𝑥 − ℎ 𝑑(ℎ + 𝑐, 𝑘) Horizontal Parabolas Right Left (𝑦 − 𝑘)2 = 4𝑐 𝑥 − ℎ 2 𝑦 (𝑦 − 𝑘)2 = −4𝑐(𝑥 − ℎ) + 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 𝑉 ℎ, 𝑘 Axis of Symmetry: 𝑦 = 𝑘 𝐹 ℎ + 𝑐, 𝑘 Focus 𝐹 ℎ − 𝑐, 𝑘 𝑥 =ℎ−𝑐 Directrix POI 𝑥 =ℎ+𝑐 𝑑 ℎ + 𝑐, 𝑘 𝑑 ℎ − 𝑐, 𝑘 focal axis and directrix 𝐅𝐨𝐜𝐚𝐥 𝐀𝐱𝐢𝐬 Vertical - UP 𝐹(ℎ, 𝑘 + 𝑐) 𝑉(ℎ, 𝑘) 𝑑(ℎ, 𝑘 − 𝑐) 𝒙=𝒉 𝒚=𝒌−𝒄 2 (𝑥 − ℎ) = 4𝑐 𝑦 − 𝑘 Vertical - DOWN 𝐅𝐨𝐜𝐚𝐥 𝐀𝐱𝐢𝐬 𝑑(ℎ, 𝑘 + 𝑐) 𝑉(ℎ, 𝑘) 𝐹(ℎ, 𝑘 − 𝑐) 2 (𝑥 − ℎ) = −4𝑐 𝑦 − 𝑘 𝒙=𝒉 𝒚=𝒌+𝒄 Vertical - GENERAL Up Down (𝑥 − ℎ)2 = 4𝑐 𝑦 − 𝑘 (𝑥 − ℎ)2 = −4𝑐(𝑦 − 𝑘) 2 𝑥 + 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 𝑉 ℎ, 𝑘 Axis of Symmetry: 𝑥 = ℎ 𝐹 ℎ, 𝑘 − 𝑐 𝐹 ℎ, 𝑘 + 𝑐 Focus 𝑦 =𝑘−𝑐 𝑑 ℎ, 𝑘 − 𝑐 Directrix 𝑦 =𝑘+𝑐 𝑑 ℎ, 𝑘 + 𝑐 General Form Difference 2 𝑥 + 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 Vertical OR Horizontal 2 𝑦 + 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 Capital C in this case is different from lower case before General → Standard 2 𝑦 + 8𝑥 + 6𝑦 − 23 = 0 complete the square!  Gather second degree variable on L.S. and first degree on R.S. 2 𝑦 + 6𝑦+9 = −8𝑥 + 23+ 9 L.S. COMPLETE SQUARE 2 (𝑦 + 3) = −8𝑥 + 32 R.S. GREATEST COMMON FACTOR (𝑦 + 2 3) = −8(𝑥 − 4) Try It 2 2𝑥 + 24𝑥 + 36𝑦 − 288 = 0 Homework • #21a, 22dg, 23b Homework • #24d