Converting Quadratic Equations into Vertex Form for Simplified Calculations and Analysis

Vertex form

The vertex form is a way to represent a quadratic function in the form of f(x) = a(x – h)^2 + k, where (h, k) represents the vertex of the parabola

The vertex form is a way to represent a quadratic function in the form of f(x) = a(x – h)^2 + k, where (h, k) represents the vertex of the parabola. In this form, “a” is the coefficient that determines the direction and width of the parabola, while (h, k) represents the horizontal and vertical translations of the parabola.

The vertex form is useful in various situations, such as finding the vertex of a parabola, determining the maximum or minimum value of the function, and graphing quadratic equations.

To convert a quadratic equation into vertex form, you can follow these steps:

1. Start with a quadratic equation in standard form: f(x) = ax^2 + bx + c.

2. Complete the square by adding and subtracting the square of half of the coefficient of the x-term. This step aims to create a perfect square trinomial.

f(x) = a(x^2 + (b/a)x) + c

Since we want to create a perfect square trinomial, we need to add and subtract (b/2a)^2:

f(x) = a(x^2 + (b/a)x + (b/2a)^2 – (b/2a)^2) + c

f(x) = a(x^2 + (b/a)x + (b/2a)^2) – a(b/2a)^2 + c

f(x) = a(x + b/2a)^2 – (ab/4a^2) + c

f(x) = a(x + b/2a)^2 + (c – (ab/4a^2))

3. Simplify the equation by combining like terms:

f(x) = a(x + b/2a)^2 + (4ac – ab^2)/4a^2

So, the equation is now in vertex form.

4. Identify the vertex coordinates. The vertex of the parabola can be found by extracting the values of (h, k) from the equation in vertex form. The vertex is represented as (h, k), where h = -b/2a and k = (4ac – ab^2)/4a^2.

Converting the quadratic equation into vertex form can help simplify calculations and provide a clear understanding of the behavior of the quadratic function.

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