Identify each equation without completing the square.
$y^2-4x-4y=0$

Verified step by step guidance

1. Start by rearranging the given equation to group the terms involving y together: \( y^2 - 4y - 4x = 0 \).
2. Notice that the equation is quadratic in terms of y. This means it can be expressed in the standard form of a quadratic equation: \( Ay^2 + By + C = 0 \).
3. Identify the coefficients for the quadratic terms: Here, \( A = 1 \), \( B = -4 \), and \( C = -4x \).
4. Recognize that this equation represents a parabola because it is a quadratic equation in y. The presence of the \( x \) term indicates that the parabola is oriented in the xy-plane.
5. To further analyze the equation, consider rewriting it in a form that highlights its structure, such as \( y^2 - 4y = 4x \), which can help in identifying the vertex and axis of symmetry of the parabola.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically in the form ax^2 + bx + c = 0, where a, b, and c are constants. In the given equation, y^2 - 4y - 4x = 0, the variable y is squared, indicating it is a quadratic in y. Understanding the structure of quadratic equations is essential for identifying their properties and solutions.

Standard Form of a Quadratic

The standard form of a quadratic equation is expressed as y = ax^2 + bx + c. This form allows for easy identification of the coefficients and the vertex of the parabola represented by the equation. In the context of the given equation, recognizing how to rearrange it into standard form is crucial for analysis and graphing.

Graphing Quadratics

Graphing a quadratic equation involves plotting a parabola on the coordinate plane. The shape and position of the parabola are determined by the coefficients of the equation. Understanding how to identify the vertex, axis of symmetry, and intercepts is vital for accurately graphing the equation derived from y^2 - 4x - 4y = 0.