Determine the vertex form and the maximum or minimum value of the function: f(x) = 2x^2 + 8x + 3

Answer 1

To determine the vertex form of the function and the maximum/minimum value, we first need to rewrite the given function in vertex form.

The vertex form of a quadratic function is given by: f(x) = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola.

Given function: f(x) = 2x^2 + 8x + 3

To rewrite it in vertex form, we need to complete the square.

f(x) = 2(x^2 + 4x) + 3
f(x) = 2(x^2 + 4x + 4) + 3 - 2(4)
f(x) = 2(x + 2)^2 - 8 + 3
f(x) = 2(x + 2)^2 - 5

Therefore, the vertex form of the function is f(x) = 2(x + 2)^2 - 5.

The vertex of the parabola is at the point (-2, -5).

Since the coefficient of x^2 is positive (a = 2), the parabola opens upward and the vertex represents the minimum value of the function.

Therefore, the minimum value of the function f(x) = 2x^2 + 8x + 3 is -5 at the point (-2, -5).