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To understand the equation of a parabola, we need to first grasp its basic form. For parabolas that open either to the left or right along the x-axis, the standard form is given by: \[ y^2 = 4ax \]Here, 'a' represents the distance from the vertex of the parabola to its focus. This relationship helps in setting the stage to derive other variations of the equation. If the vertex is shifted from its original location, this basic equation can be modified to accommodate the shift, which brings us to our next concept: transforming the parabola equation.

To solve the problem, we need to understand the relationship between the vertex and the focus of a parabola. If the focus is at the origin \((0, 0)\) and the vertex is not at the origin but rather shifted to \((-k, 0)\), then the configuration changes the equation. This transformation means that the vertex is k units to the left of the focus on the x-axis. To incorporate this shift into our equation, we need to adjust for the vertex's new position. This goes from the standard form to:\[ (y - 0)^2 = 4a(x + k) \]Which translates to \((y)^2 = 4k(x + k)\).Finally, expanding this you get:\[ y^2 = 4kx + 4k^2\]This derived equation fits our original scenario, with the focus at the origin and the vertex moved to \((-k, 0)\).

Transforming the parabola means shifting its vertex while maintaining the focus. For the given exercise, the vertex is at \((-k, 0)\) while the focus remains at \((0, 0)\). The transformation modifies the standard equation \((y^2 = 4ax)\) into a new form. Hence, the transformation process involves:

• Recognizing the vertex shift
• Adjusting the equation accordingly
• Expanding and simplifying

So, starting with the standard form and incorporating the shift, we get the modified equation: \[(y - 0)^2 = 4a(x + k)\]. Further simplification yields similar terms added to the original equation, converting it to: \[ y^2 = 4kx + 4k^2 \] This derived equation expresses a parabola whose focus is at the origin and the vertex is shifted horizontally.