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Completing the square is a method used to simplify quadratic equations by turning part of the equation into a perfect square trinomial. This method is useful when transforming the equation of a parabola into its standard form. To understand completing the square, follow these simple steps:
First, isolate the quadratic and linear terms. For example, from the equation given: \[ y^{2} + 6x + 10y + 19 = 0 \]
Separate terms with y:
\[ y^{2} + 10y = -6x - 19 \]
Next, take half of the coefficient of y, square it, and add/subtract it back to the equation:
\[ \text{Coefficient of } y = 10, \text{Half of } 10 = 5, 5^{2} = 25 \]
So, add and subtract 25:
\[ y^{2} + 10y + 25 - 25 = -6x - 19 \]
This converts perfectly to:
\[ (y + 5)^{2} - 25 = -6x - 19 \]
Finally, rearrange the terms to form:
\[ (y + 5)^{2} = -6(x - 1) \]
Completing the square helps us identify key features of the parabola, such as its vertex and focus.

The standard form of a parabola offers a clear view of key characteristics like the vertex and the direction it opens. For a vertical parabola, the standard form is:
\[ (y - k)^{2} = 4p(x - h) \]
In this equation:

• (h, k) is the vertex of the parabola.
• p is the distance from the vertex to the focus.

From our given parabola:
\[ (y + 5)^{2} = -6(x - 1) \]
By comparing, we see:

• h = 1
• k = -5
• 4p = -6

From this, solve for p:
\[ 4p = -6 \]\[ p = -\frac{3}{2} \]
So the standard form tells us the vertex is at (1, -5) and the parabola opens to the left because p is negative.

The focus and directrix of a parabola give us insights into its geometric properties. The focus is a specific point that lies inside the parabola, serving as a point from which distances are measured that define the curve. The directrix is a line outside the parabola used in the same way.

• For the form \( (y - k)^{2} = 4p(x - h) \)
• Focus: Located at \( (h + p, k) \)
• Directrix: Line given by \( x = h - p \)

For the given standard form:
\[ (y + 5)^{2} = -6(x - 1) \]
The focus point is:
\[ (h + p, k) = \left(1 - \frac{3}{2}, -5\right) = \left(-\frac{1}{2}, -5\right) \]
The directrix line is:
\[ x = h - p = 1 + \frac{3}{2} = \frac{5}{2} \]
Understanding the focus and directrix helps sketch the parabola and visualize its shape and direction.

The parabola axis, also known as the axis of symmetry, is a line that divides the parabola into two mirror-image halves. This axis always goes through the vertex and is perpendicular to the direction in which the parabola opens.

• For vertical parabolas: The axis is a vertical line.
• For horizontal parabolas: The axis is a horizontal line.

Given the standard form of our parabola:
\[ (y + 5)^{2} = -6(x - 1) \]
The axis of symmetry passes through the vertex at (1, -5). Since our parabola opens horizontally, the axis of symmetry is a horizontal line. Thus, the equation of the axis is:
\[ y = -5 \]
The axis helps us to accurately draw and understand the parabola’s orientation and properties.