conic section - hyperbola

• The standard form of a Hyperbola is represented as 𝑥²��² − 𝑦²𝑏² = 1.
• The transverse axis of a Hyperbola is when 𝑦 = 0 and 𝑥 = 0.
• The conjugate axis of a Hyperbola is when 𝑥 = 0 and 𝑦 = 0.
• The vertices of a Hyperbola are (𝑎, 0) and (−𝑎, 0).
• Simplify and rearrange as needed to match the pattern.
• For each set (𝑥 and 𝑦 separately), take the number in front of the first degree term, divide it by 2 and square it.
• Draw extended diagonals of the rectangle.
• Draw the hyperbola with extended diagonals as asymptotes and vertices at (ℎ+𝑎, 𝑘) and (ℎ−𝑎, 𝑘).
• Factor both trinomials and rewrite them.
• Factor out the coefficient of the 𝑥2 term and the ��2 term.
• Simplify the right hand side and divide each term by that number so that the right hand side equals 1.
• Let 𝑐2 = ��2 + ��2.
• Rearrange the terms so that the y terms are together and the x terms are together.
• Horizontal and vertical foci are (ℎ+𝑐, 𝑘) and (ℎ−𝑐, 𝑘).
• Add both numbers' values to both sides of the equation.
• Plot rectangle with corners (ℎ−𝑎, 𝑘+𝑏), (ℎ+𝑎, 𝑘+𝑏), (ℎ−𝑎, 𝑘−𝑏), (ℎ+𝑎, 𝑘−𝑏).
• Arrange the equation so all the terms with variables are on the same side and the constant is on the other side.
• The asymptotes of a Hyperbola are �� = ���� (�� − ℎ) + 𝑘 and 𝑦 = −𝑏�� (�� − ℎ) + 𝑘.
• Plot the vertices (ℎ + ��, 𝑘) and (ℎ − 𝑎, 𝑘).
• Plot rectangle with corners (𝑎, 𝑏), (𝑎, −𝑏), (−𝑎, 𝑏), and (−𝑎, −𝑏).
• Plot the extended diagonals of the rectangle.
• Plot the hyperbola with extended diagonals as asymptotes and vertices at (��, 0) and (−𝑎, 0).
• Plot rectangle with corners ( ℎ − 𝑎 , 𝑘 + 𝑏 ) , ( ℎ + 𝑎 , 𝑘 + 𝑏 ) , ( ℎ − 𝑎 , 𝑘 − 𝑏 ) and ( ℎ + 𝑎 , 𝑘 − 𝑏 )
• Graph 𝑦 = ���� (�� − ℎ) + 𝑘 and 𝑦 = −𝑏�� (𝑥 − ℎ) + 𝑘.
• Express into standard form.
• Plot the hyperbola with asymptotes and vertices as guides.
• Plot rectangle with corners (ℎ − ��, 𝑘 + 𝑏), (ℎ + 𝑎, �� + 𝑏), (ℎ − 𝑎, 𝑘 − 𝑏), and (ℎ + 𝑎, 𝑘 − 𝑏).
• The slope method for finding the standard form of a Hyperbola is to express into standard form, graph �� = 𝑏𝑎𝑥 and 𝑦 = −𝑏𝑎𝑥, plot the vertices (𝑎, 0) and (−𝑎, 0), and draw the hyperbola with asymptotes and vertices as guides.
• The central rectangle method for finding the standard form of a Hyperbola is to express into standard form, graph �� = 𝑏𝑎𝑥 and 𝑦 = −𝑏𝑎𝑥, plot the vertices (0, ��) and (0, −𝑏), and draw the hyperbola with asymptotes and vertices as guides.