Finding the Equation of a Parabola Given Vertex and Y-Intercept

Finding the equation of a parabola can be a common task in algebra. In this article, we will explore how to determine the equation of a parabola when provided with its vertex and y-intercept. We will cover multiple methods and derive the final equation step-by-step.

Finding the Equation of the Parabola Given the Vertex and Y-Intercept

Method 1: Using the General Form

Consider a parabola in its general form:

[y ax^2 bx c]

We are given that the vertex is at point (2, -3) and the y-intercept is 5. The vertex form of a parabola is given by:

[y - k a(x - h)^2]

Substituting the vertex (2, -3) into the vertex form, we get:

[y 3 a(x - 2)^2]

We are also given a y-intercept of 5. Therefore, substituting (0, 5) into the equation, we have:

[5 3 a(0 - 2)^2]

[8 4a] [a 2]

Substituting the value of a back into the vertex form, we get:

[y 3 2(x - 2)^2]

Expanding the equation:

[y 3 2(x^2 - 4x 4)] [y 3 2x^2 - 8x 8] [y 2x^2 - 8x 5]

Method 2: Using the Standard Form

Consider the standard form of a parabola:

[y ax^2 bx c]

We are given that the vertex is at (2, -3), which means:

[-frac{b}{2a} 2] [b -4a]

Also, we know the y-intercept is 5, which means the point (0, 5) is on the parabola:

[5 a(0)^2 - 4a(0) c] [c 5]

We also have the point (2, -3) on the parabola:

[-3 4a - 8a c] [-3 -4a 5] [-8 -4a] [a 2] [b -8]

Therefore, the equation of the parabola is:

[y 2x^2 - 8x 5]

Method 3: Using a Special Form

Consider the special form of a parabola:

[x ay^2 by c]

We are given the same conditions: vertex (-2, 3) and y-intercept (0, 5).

From the vertex form:

[x 2 a(y - 3)^2]

Substituting (0, 5) into the equation:

[0 2 a(5 - 3)^2] [2 4a] [a frac{1}{2}] [b -3] [c frac{5}{2}]

Therefore, the equation of the parabola is:

[x frac{1}{2}y^2 - 3y frac{5}{2}]

Conclusion

In this article, we have explored three methods to find the equation of a parabola given its vertex and y-intercept. The equations derived are:

[y 2x^2 - 8x 5] [x frac{1}{2}y^2 - 3y frac{5}{2}]

These methods involve the general form, vertex form, and a special form of the parabola equation. Each method provides a unique approach to solving the problem, demonstrating the versatility of parabolic equations in mathematics.

For a visual representation, please refer to the Desmos graph below:

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