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. $x^{2} - y^{2} - 2 x - 4 y - 4 = 0$

Name: College Algebra
Author: Blitzer
ISBN: 9780136970514

검증된 단계별 안내

Start with the given equation: $x^2 - y^2 - 2x - 4y - 4 = 0$.

Group the $x$ terms and $y$ terms together: $(x^2 - 2x) - (y^2 + 4y) = 4$ (move the constant to the right side).

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.

Rewrite the equation with completed squares: $(x^2 - 2x + 1) - (y^2 + 4y + 4) = 4 + 1 - 4$.

Express the perfect square trinomials as binomials squared: $(x - 1)^2 - (y + 2)^2 = \text{(simplified right side)}$. 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.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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. x2−y2−2x−4y−4=0x^2−y^2−2x−4y−4=0

주요 개념

Completing the Square

Standard Form of a Hyperbola

Foci and Asymptotes of a Hyperbola

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. $x^{2} - y^{2} - 2 x - 4 y - 4 = 0$