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The vertex form of a parabola is a very handy way to express the equation of the parabola. It focuses on the vertex, which is a significant point since it represents the tip of the curve. When we talk about a parabola's vertex at the origin, it makes things simpler, as the equation reduces.
To express a parabola with its vertex at the origin

• The general vertex form is written as \( y = ax^2 \) for a parabola that opens upwards or downwards (vertical axis).
• If the parabola opens to the left or right (horizontal axis), the form would be \( x = ay^2 \).

For our given problem, we assume the parabola opens either up or down since the focus is on the y-axis. Thus, we use \( y = ax^2 \), where \( a \) is a constant that influences the parabolic stretch and direction.
To find \( a \), we use the condition that the curve passes through a particular point, resulting in a straightforward substitution to solve for this unknown.

The focus of a parabola is one of its defining characteristics. It's a point inside the parabola, which, together with the directrix, defines the shape of the parabola.
For parabolas with a vertex at the origin and symmetry about the y-axis:

• The standard equation of a parabola is expressed as \( y = \frac{1}{4c}x^2 \).
• Here, \( c \) is the distance from the vertex to the focus.
• The focus is given by the coordinates \((0, c)\).

For our specific problem:

• Since the value of \( a \) is obtained as \( -\frac{10}{49} \), it equates to \( \frac{1}{4c} \).
• We can derive \( c \) by solving \( a = \frac{1}{4c} \), providing \( c = -\frac{49}{40} \).

By identifying the focus at \((0, -\frac{49}{40})\), it confirms the geometry adheres to the assumption and the required condition.

Coordinate geometry allows us to understand and represent geometric shapes within a coordinate plane. It is exceptionally useful for analyzing the properties and relationships of a parabola.
In this context, to find a parabola's equation:

• Start with identifying key points such as the vertex and the focus.
• Utilize given points like \((7, -10)\) to substitute into the vertex form equation \( y = ax^2 \).
• Solving \(-10 = 49a\), we find \( a = -\frac{10}{49} \), linking the specific coordinate conditions with the algebraic expression.

Moreover, coordinate geometry provides a framework to draw the parabola accurately, verifying aspects like the width, stretch, and orientation based on calculations derived from the vertex form and other parabolic traits like focus.