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Completing the square is a method used to transform a quadratic equation into a form that is easier to analyze or graph. In this process, we focus on the quadratic expression, which typically involves a term with a squared variable and a linear term. To complete the square:

• Identify the coefficient of the linear term.
• Take half of this coefficient and square it.
• Add and subtract this squared value within the expression.

For instance, consider the expression \(y^2 + 5y\). The coefficient of \(y\) is 5. Halving it gives \(\frac{5}{2}\), and squaring this value produces \(\left(\frac{5}{2}\right)^2 = \frac{25}{4}\). By adding and subtracting \(\frac{25}{4}\) within the parenthesis, we rewrite the quadratic part as a perfect square trinomial: \((y + \frac{5}{2})^2\). This helps in transforming the original quadratic expression into a more manageable form, making graphing much simpler.

The vertex form of a parabola is a way of expressing the quadratic equation to highlight its vertex, which is its highest or lowest point, depending on direction. The general form is \(x = a(y - k)^2 + h\) or \(y = a(x - h)^2 + k\), where

• \((h, k)\) represents the vertex.
• \(a\) determines the direction and width of the parabola.

In the equation provided, \(x = 5(y + \frac{5}{2})^2 + \frac{115}{4}\), it fits the vertex format \(x = a(y - k)^2 + h\):

• The vertex \((h,k)\) is \(\left(\frac{115}{4}, -\frac{5}{2}\right)\).
• \(a = 5\) causes the parabola to open to the right, since it's greater than zero.

Having the equation in vertex form provides a straightforward way to quickly identify key features of the parabola, which is crucial for graphing.

Graphing quadratic equations involves plotting a curve that represents the solutions to the equation. For a parabola, the graph is a smooth, symmetrical curve. Here’s how to graph using the vertex form:

• Start by locating the vertex on your graph, \((\frac{115}{4}, -\frac{5}{2})\), which acts as the "starting" or "turning" point.
• Since our parabola opens to the right, it will extend outward from the vertex.
• Select additional \(y\) values around the vertex to find corresponding \(x\) values using \(x = 5(y + \frac{5}{2})^2 + \frac{115}{4}\).

Plot these additional points and draw a smooth curve through them. Ensure the parabola is symmetrical around its axis, which is a vertical line passing through the vertex in this case. This method not only accurately reflects the shape of the quadratic function but also leverages the vertex form to simplify the drawing process.