91Ó°ÊÓ

The vertex of a parabola is a critical point that indicates the position of the parabola's minimum or maximum value. In our specific problem, the parabola's equation is given as \( x = -2(y + 3)^2 \). This equation is structured in the standard form \( x = a(y - k)^2 + h \), where the vertex is represented by \((h, k)\).
For the given equation \( x = -2(y + 3)^2 \), by comparing it to the standard form, we can see that the vertex is located at the point \((0, -3)\).
This point tells us where the parabola turns and starts opening to one side. Because the coefficient \(a\) is negative in this case \((a = -2)\), this parabola opens to the left, with the vertex sitting on the boundary.

The axis of symmetry is a line that runs through the vertex and divides the parabola into two mirror-image halves. For vertically opening parabolas, this line is vertical, while for horizontal parabolas like ours, it's horizontal.
Our equation \( x = -2(y + 3)^2 \) uncovers the axis of symmetry as being \( y = -3 \), which is aligned horizontally through the vertex \((0, -3)\).
It's helpful to think of the axis of symmetry as a balanced centerline, ensuring the shape is symmetrical.

Understanding the domain and range of a parabola can help in sketching the graph accurately. The domain specifies all the possible x-values the parabola can have, whereas the range specifies all the possible y-values.
In the given parabola \( x = -2(y + 3)^2 \), the sideways opening implies that the domain is restricted on one end.

• **Domain:** For our equation, the parabola opens left. The domain is \((-ty, 0]\). This means x-values range from negative infinity to 0.
• **Range:** Our sideways parabola accepts all possible y-values. Thus, the range is \((-ty, ty)\).

Graphing a parabola by hand can be simplified through understanding the relations and equations involved. With our equation \( x = -2(y + 3)^2 \), follow these easy steps to make the graphing process practical:

• **Plot the Vertex:** Begin by marking the vertex at \((0, -3)\) on a graph.
• **Mark the Axis of Symmetry:** Draw the horizontal line \( y = -3 \) to visualize symmetry.
• **Choose Additional Points:** Select a few convenient y-values, substitute them into the equation, and calculate corresponding x-values.
• **Draw the Curve:** Connect your points with a smooth curve. Note the opening direction is left due to the negative coefficient.

Utilizing these steps aids in creating an accurate visual representation of the parabola by hand, allowing for a comprehensive understanding before confirming with a graphing calculator.