Convert equations of hyperbolas from general to standard form

Learning Objectives Identify the general form of a hyperbola's equation. Correctly group variable terms and move the constant term in a general equation. Apply the 'completing the square' method to both the x and y variable groups. Factor the resulting perfect square trinomials accurately. Manipulate the equation to set it equal to 1, resulting in the standard form. Convert a hyperbola's equation from general form to standard form to identify its center and orientation. Distinguish between the equations for a horizontal and a vertical hyperbola. How can a messy equation like 4x² - 9y² - 16x + 18y - 29 = 0 possibly describe the elegant, curved path of a comet? ☄️ Let's find out! This tutorial will guide you through the algebraic process of transfor...

TermDefinitionExample General Form of a HyperbolaAn equation of the form Ax² + Cy² + Dx + Ey + F = 0, where A and C have opposite signs (one is positive, one is negative). This form hides the hyperbola's properties.9x² - 16y² - 36x - 32y - 124 = 0 Standard Form of a HyperbolaThe form of a hyperbola's equation that clearly reveals its center (h, k) and orientation. There are two types: one for horizontal and one for vertical hyperbolas.Horizontal: (x-2)²/16 - (y+1)²/9 = 1. Vertical: (y+1)²/9 - (x-2)²/16 = 1. Completing the SquareAn algebraic technique used to convert a binomial of the form x² + bx into a perfect square trinomial by adding (b/2)². This is the core process for converting from general to standard form.To complete the square for x² + 6x, we add (6/2)² = 3² = 9. This...

Standard Form (Horizontal Transverse Axis) \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 Use this form when the hyperbola opens left and right. The x-term is positive. The center is (h, k). Standard Form (Vertical Transverse Axis) \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 Use this form when the hyperbola opens up and down. The y-term is positive. The center is (h, k).