91Ó°ÊÓ

Completing the square is a technique to convert a quadratic equation into a perfect square trinomial. This method makes it easier to transform the equation into a form that reveals key features of the graph, such as the vertex and axis of symmetry. To complete the square for the quadratic expression in our given problem, we start with:
y^{2} - 4y + 4x + 4 = 0
First, move the constants to the other side:
y^{2} - 4y = -4x - 4
Next, to complete the square on the left-hand side, take half of the coefficient of y (which is -4), square it, and add it inside the equation:
y^{2} - 4y = (y - 2)^2 - 4
Adding this back into the equation, we get:
(y - 2)^2 - 4 = -4x - 4.
Add 4 to both sides to complete the square:
(y - 2)^2 = -4x.
Now, the quadratic equation is in a form that makes graphing straightforward.

The standard form of a parabola helps in readily identifying its key properties. The general equation of a parabola in standard form is written as:
(y - k)^2 = 4p(x - h).
Here,

• (h, k) is the vertex of the parabola
• p is the distance from the vertex to the focus
• 4p is the coefficient that determines the shape and direction of the parabola.

For our equation, (y - 2)^2 = -4x, we can identify the parameters directly:
The vertex (h, k) is (0, 2)
Since 4p = -4, we can solve for p: p = -1.
Knowing this standard form allows us to understand the orientation and the specific details such as vertex, focus, and directrix.

Graphing parabolas becomes straightforward after transforming the equation into standard form. With our standard form equation, (y - 2)^2 = -4x, we can easily graph it by following these steps:

• Plot the vertex at the point (0, 2)
• Identify the direction of the parabola. Since the coefficient of x is negative, the parabola opens to the left.
• Find the focus using the value of p, which is -1. The focus will be at (-1, 2).
• Draw the directrix, which is a vertical line at x = h - p. Hence, the directrix is the line x = 1.

Once these points are plotted, sketch the parabola opening to the left from the vertex.

Understanding the properties of parabolas is crucial for graphing and analyzing them. The key properties are:

• Vertex: The turning point of the parabola where it changes direction. For our problem, the vertex is (0, 2).
• Focus: A point inside the parabola where all reflected lines converge. The focus for our equation is (-1, 2).
• Directrix: A line outside the parabola that the curve approaches but never touches. For our example, the directrix is x = 1.
• Axis of symmetry: A line that divides the parabola into two mirror-image halves. For a horizontally oriented parabola, this line is parallel to the y-axis.
• Direction of opening: The side towards which the parabola opens. Since our equation’s coefficient is negative, the parabola opens to the left.

These properties help in recognizing and sketching the parabola accurately on a graph.