The vertex form of a parabola is a key concept in algebra and geometry, essential for understanding and manipulating the graphs of quadratic functions. When learning how to write an equation for a parabola, it’s crucial to understand the transformation between its standard form \( y = ax^2 + bx + c \) and vertex form \( y = a(x – h)^2 + k \). In this form, \( (h, k) \) represents the vertex, the pinnacle (or nadir) of the parabola, providing a straightforward way to graph and interpret the quadratic function.

## Understanding the Vertex Form

The vertex form of a parabola’s equation, \( y = a(x – h)^2 + k \), is particularly useful because it directly reveals the vertex \( (h, k) \). The vertex denotes the maximum or minimum point on the graph, depending on whether the parabola opens upwards \((a > 0)\) or downwards \((a < 0)\). This form simplifies both the plotting and transformation processes involved in manipulating parabolas.

## How to Write an Equation for a Parabola

Learning how to write an equation for a parabola in vertex form involves these steps:

- Identify the vertex \( (h, k) \) of the parabola.
- Use another point on the graph, such as the y-intercept, to solve for the coefficient \( a \).
- Substitute these values into the vertex form equation.

For example, if the vertex of the parabola is \( (2, 5) \) and another point on the parabola is \( (0, 9) \), you can substitute these into the vertex form equation to solve for \( a \). Using the steps outlined, you can plug in these values and simplify the equation accordingly.

## From Vertex Form to Standard Form

Sometimes it is necessary to convert the vertex form of the equation into its standard form. To do this, expand the squared term and simplify:

\[ y = a(x – h)^2 + k \]

Expanding yields:

\[ y = a(x^2 – 2hx + h^2) + k \]

Simplify to obtain:

\[ y = ax^2 – 2ahx + ah^2 + k \]

This provides the standard form of the quadratic equation, \( y = ax^2 + bx + c \).

## Applications and Examples

Consider a parabola with the vertex at \( (2, 5) \) and passing through the y-intercept at \( (0, 9) \). Plugging these points into the vertex form equation and solving for \( a \) gives you the specific quadratic function representing this parabola. After finding \( a \), you can write the equation in vertex form and, if needed, convert it to standard form for further analysis.

## Summary

Understanding how to write an equation for a parabola in vertex form is an essential skill in algebra, enabling you to easily graph and interpret parabolas. Whether starting from the vertex and finding \( a \) or converting between vertex and standard forms, mastering these techniques is fundamental to solving quadratic equations efficiently and accurately.

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