Convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $x^2-y^2-2x-4y-4=0$

Verified step by step guidance

1. Start with the given equation: \(x^2 - y^2 - 2x - 4y - 4 = 0\).
2. Group the \(x\) terms and \(y\) terms together: \((x^2 - 2x) - (y^2 + 4y) = 4\) (move the constant to the right side).
3. Complete the square for the \(x\) terms: take half of \(-2\), which is \(-1\), square it to get \$1\(, and add it inside the parentheses. Do the same for the \)y\( terms: half of \)4\( is \)2\(, square it to get \)4$, and add it inside the parentheses. Remember to balance the equation by adding these values to the right side as well.
4. Rewrite the equation with completed squares: \((x^2 - 2x + 1) - (y^2 + 4y + 4) = 4 + 1 - 4\).
5. Express the perfect square trinomials as binomials squared: \((x - 1)^2 - (y + 2)^2 = ext{(simplified right side)}\). This is the standard form of a hyperbola. From here, identify the center, vertices, foci, and write the equations of the asymptotes based on the standard form.

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)² or (y - k)² by adding and subtracting terms. This technique helps convert the given equation into a recognizable conic section form, making it easier to analyze and graph.

Standard Form of a Hyperbola

The standard form of a hyperbola is (x - h)²/a² - (y - k)²/b² = 1 or its vertical counterpart. Writing the equation in this form reveals the center (h, k), the orientation, and the values of a and b, which are essential for graphing and understanding the hyperbola's shape.

Foci and Asymptotes of a Hyperbola

The foci are two fixed points that define the hyperbola, located along the transverse axis at a distance c from the center, where c² = a² + b². Asymptotes are lines that the hyperbola approaches but never touches, with slopes ±b/a or ±a/b depending on orientation, guiding the graph's shape.