Quadratic Functions in Vertex Form

Vertex Form of a Quadratic Function

A quadratic function is a polynomial function of degree two, represented in general form as:

[f(x) = ax^2 + bx + c]

where (a, b,) and (c) are real numbers and (a eq 0). The vertex form of a quadratic function is another way to represent the function and is particularly useful for identifying the vertex of the parabola. The vertex form is given by:

[f(x) = a(x - h)^2 + k]

where ((h, k)) is the vertex.

Vertex of a Quadratic Function

The vertex of a quadratic function represents the maximum or minimum value of the function. It is the point where the parabola changes direction, going from decreasing to increasing or vice versa. The vertex is located on the axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves.

The vertex ((h, k)) is calculated using the following formulas:

[h = -\frac{b}{2a}] [k = f(h) = f\left(\frac{-b}{2a}\right)]

Writing a Quadratic Function in Vertex Form

To find the vertex form of a quadratic function, follow these steps:

1. Identify the coefficients (a, b,) and (c).
2. Calculate (h) using the formula (h = -\frac{b}{2a}).
3. Evaluate (k) by plugging (h) into the function: (k = f(h) = f\left(\frac{-b}{2a}\right)).

Converting General Form to Vertex Form

To convert a quadratic function in general form to vertex form, follow these steps:

1. Identify the coefficients (a, b,) and (c).
2. Calculate (h) using the formula (h = -\frac{b}{2a}).
3. Evaluate (k) by plugging (h) into the function: (k = f(h) = f\left(\frac{-b}{2a}\right)).
4. The vertex form of the function is: (f(x) = a(x - h)^2 + k).

Example

Consider the quadratic function:

[f(x) = 3x^2 - 6x + 7]

In general form, this function is:

[f(x) = 3x^2 - 6x + 7]

To find the vertex form, we need to find the coefficients (a, b,) and (c):

[a = 3, b = -6, c = 7]

Now, we can find (h) and (k):

[h = -\frac{b}{2a} = -\frac{-6}{2(3)} = \frac{6}{6} = 1] [k = f(h) = f(1) = 3(1)^2 - 6(1) + 7 = 3 - 6 + 7 = 6]

The vertex form of the function is:

[f(x) = 3(x - 1)^2 + 6]

In this form, we can see that the vertex of the parabola is ((1, 6)), and the parabola opens upwards, as (a eq 3).