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Completing the square is a technique used in algebra to transform a quadratic equation into a specific form. This is particularly helpful for identifying features of conic sections, like the vertex in a parabola. Let's break it down:- **Identify the quadratic part**: Look at the given equation and focus on the quadratic terms. For instance, in the equation, \( y^2 + 4y \), concentrate on the expression involving \( y \).- **Take half of the linear coefficient**: Divide the linear coefficient (the coefficient of \(y\), which is 4) by 2, giving 2.- **Square it**: Now, square the result (2), which results in 4.- **Adjust the equation**: Add this square (4) into the expression, and to keep the equation balanced, subtract it as well.By completing the square, you convert \( y^2 + 4y \) into \((y + 2)^2 - 4\). This makes it easier to manipulate the equation and identify the structure of the graph, such as turning the equation \( x = y^2 + 4y - 1 \) into \( x = (y + 2)^2 - 5 \). This completed square form reveals more about the graph's characteristics.

The vertex form of a parabola is a useful representation that directly shows the vertex, an essential feature when graphing. A parabola can usually be expressed in the form:\[ x = a(y-k)^2 + h \]In this form:- **\(a\)**: Determines the direction and width of the parabola. For sideways parabolas, if \(a\) is positive, the graph opens to the right, and if negative, it opens to the left.- **\((h, k)\)**: Denotes the coordinates of the vertex, a pivotal point of the parabola.In our example, \(x = (y + 2)^2 - 5\), the vertex is clear at the point \((-5, -2)\). The form helps to quickly pinpoint the vertex and gives an overview of the parabola's direction. Being in vertex form makes graphing and analyzing the parabola more straightforward.

Conic sections arise from intersecting a plane with a cone. Depending on how the plane cuts through the cone, we can get different shapes:- **Circle**: Occurs when the cutting plane is perpendicular to the cone's axis, resulting in a shape where all points are equidistant from a center.- **Ellipse**: When the plane cuts through at an angle but doesn’t reach the base, leading to an elongated round shape.- **Parabola**: This is the result of the plane being parallel to the slant of the cone, resulting in a U-shaped curve.- **Hyperbola**: When the plane cuts through both halves of the cone, forming two open curves.In the given problem, finding the type of conic section involved was simplified by knowing that only \( y \) is squared, and not \( x \). This single squared term directly indicates a parabola, differentiating it from other conic sections which would involve squared terms of both variables.

Graphing a parabola or any conic section accurately requires understanding its structure and distinct features. Here is a step-by-step guide to ensuring a great graph:- **Identify the vertex**: Start by determining the vertex using the vertex form. This gives you a central point from which the graph will expand.- **Symmetry**: Parabolas are symmetric about their line of symmetry, which for equations in the form \(x = (y - k)^2 + h\) is the line parallel to the x-axis.- **Plot additional points**: To draw a detailed curve, select several \(y\) values, calculate their corresponding \(x\), and plot these points.- **Sketch the shape**: Use the points to sketch a smooth curve. Ensure the curve opens in the right direction, matching the sign and coefficient of the squared term.In our exercise, after finding the vertex at \((-5, -2)\) and understanding it opens to the right, you can confidently draft the parabola across a grid. Always check for symmetry and additional points to ensure the sketch is accurate.