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Understanding the vertex form of a parabola is essential to write an equation accurately. The vertex form of a parabola is expressed as \( y = a(x-h)^2 + k \), where \((h, k)\) represents the vertex of the parabola.
The vertex serves as a critical turning point on the graph, where the parabola changes direction.
In our specific example, the vertex is \((-3, -1)\). The formula provides a structured way to incorporate this important point directly into the equation.

• \( h \) translates the graph horizontally, affecting its left or right movement.
• \( k \) translates the graph vertically, impacting its up or down positioning.

Using the vertex form makes it easier to graph the parabola or alter it for different contexts. With respect to our example, substituting the vertex into the vertex form gives us: \( y = a(x+3)^2 -1 \). This equation now articulates the parabola's position in relation to its center.

The value of 'a' in the vertex form of a parabola plays a crucial role in determining its shape and direction.
It's what impacts the parabola's width and whether it opens upwards or downwards.
To find and elucidate the value of \( a \), you can use any point through which the parabola passes, aside from the vertex.
In our worked example, this is the point \((-2, -3)\).

• This involves substituting the known \( x \) and \( y \) values from the point into the equation \( y = a(x-h)^2 + k \).
• Once substituted, the only unknown in the equation is \( a \), making it simple to solve for its value.

In our case: \[ -3 = a(-2+3)^2 -1 \]By simplifying, you find \( a = -2 \).
This negative value tells us that the parabola opens downwards, offering an inverted U-shape.

Once you've determined the parameters of your parabola, writing its equation in standard form is straightforward. This form allows you to clearly express the trajectory and curvature of a parabola. The formula to follow is:
\( y = a(x-h)^2 + k \).
Having determined \( a \), \( h \), and \( k \), each value fits seamlessly into the standard form. Thus, the final equation represents the complete geometric representation of your parabola.

• From the vertex, \( h = -3 \) and \( k = -1 \).
• From the secondary point, we've derived that \( a = -2 \).

Now, place these values into the formula: \[ y = -2(x+3)^2 -1 \]This expression is the standard form equation of the parabola passing through specified points with defined peaks and orientation. Writing parabolic equations in such a format makes analysis and graph plotting more intuitive and descriptive.