Finding the Equation of a Parabola with a Given Vertex

Parabolas are fundamental conic sections that have a wide range of applications in mathematics, physics, and engineering. In this article, we will explore how to find the equation of a parabola given its vertex and two points through which it passes. This process involves the use of the standard form and vertex form of a quadratic equation. Letrsquo;s delve into the detailed steps to determine the equation of the parabola and understand the underlying concepts.

Standard Form and Vertex Form

The standard form of a parabola equation that opens upwards or downwards is:

y ax^2

When the vertex of the parabola is at the origin (0,0), the equation simplifies further as it lacks linear and constant terms:

y ax^2

Identifying the Parabolarsquo;s Equations

Given a vertex at (0,0) and points (2,3) and (-2,3), we need to determine the value of a in the equation y ax^2 that makes the parabola pass through these points. Letrsquo;s solve for a.

1. **Substituting Point (2,3):**

3 a(2)^2

3 4a

a frac{3}{4}

The equation of the parabola is:

y frac{3}{4}x^2

To verify, substitute the point (-2,3) into the equation:

y frac{3}{4}(-2)^2 frac{3}{4} cdot 4 3

The equation satisfies both points (2,3) and (-2,3). Hence, the final equation of the parabola is:

y frac{3}{4}x^2

Special Properties and General Form

The parabola has the y-axis as its axis of symmetry. Its equation can be written in the standard form as:

x^2 4ay

Substituting the coordinates (2,3) into this equation:

(2)^2 4a cdot 3

4 12a rightarrow a frac{1}{3}

The general form of the parabola is:

x^2 4 cdot frac{1}{3} cdot y rightarrow x^2 frac{4}{3}y

Summary and Key Points

For a parabola in standard form, the equation can also be expressed as:

3x^2 - 4y 0

The vertex form y a(x-h)^2 k helps to identify the vertex directly. Plugging in the point (2,3) with the vertex at (0,0):

3 a(2)^2 0 rightarrow 4a 3 rightarrow a frac{3}{4}

Thus, the equation of the parabola is:

y frac{3}{4}x^2

Understanding the properties and forms of a parabola is essential for various applications, including optics, design, and solving quadratic equations. The steps described here can be applied to find the equation of any parabola given its vertex and points through which it passes.