using the completing-the-square method, find the vertex of the function f(x)=5x^2+10x+8 and indicate whether it is a minimum or a maximum and at what point
To find the vertex of the function using the completing-the-square method, we need to rewrite the function in the form of:
f(x) = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
For the function f(x) = 5x^2 + 10x + 8, we start by factoring out the common factor of 5 from the first two terms:
f(x) = 5(x^2 + 2x) + 8
Next, we need to complete the square for the quadratic term (x^2 + 2x). To do so, we take half of the coefficient of the linear term (2x), square it, and add/subtract it inside the parentheses:
f(x) = 5(x^2 + 2x + (2/2)^2) + 8 - 5(2/2)^2
f(x) = 5(x^2 + 2x + 1) + 8 - 5(1)
f(x) = 5(x^2 + 2x + 1) + 8 - 5
f(x) = 5(x + 1)^2 + 3
Now the function is in the desired form, and we can see that the vertex is at the point (-1, 3).
Since the coefficient of the squared term (5) is positive, the parabola opens upwards. Therefore, the vertex (-1, 3) is the minimum point of the function f(x) = 5x^2 + 10x + 8.