Final answer:
The vertex of the function 4x^2 + 4x + 1 represents a minimum point because the parabola opens upwards due to a positive coefficient of x^2. By using the formula for the vertex, we find it to be at (-1/2, 1), which indicates the minimum value of the function is 1.
Step-by-step explanation:
The function mentioned in the question is 4x squared + 4x + 1, which we can rewrite as 4x^2 + 4x + 1.
Because the coefficient of the x^2 term is positive (4), the parabola opens upwards, making the vertex a minimum point on the graph.
To determine the exact coordinates of the vertex, we can use the vertex formula or complete the square.
To find the vertex, we first find the x-coordinate using the formula -b/(2a), where a and b are the coefficients from the quadratic term and linear term, respectively.
In this case, a = 4 and b = 4, hence the x-coordinate is -4/(2*4), which simplifies to -1/2.
To find the y-coordinate, we substitute this value back into the original equation: 4(-1/2)^2 + 4(-1/2) + 1, which simplifies to 1.
So, the vertex is at the point (-1/2, 1), and since this is a minimum, the smallest value of the function is 1