For a general quadratic equation, we want to find the equations for the vertex (h, k).
The values of the vertex are:
- h = -b/(2*a)
- k = f(h) = b^2/(4a) - b^2/(2a) + c
We start with the general quadratic equation:
f(x) = a*x^2 + b*x + c
To find the x-value of the vertex (h in this case) we need to find the zero of the first derivate of f(x) (because the vertex is a minimum/maximum of the function).
We have:
f'(x) = 2*a*x + b
We solve:
f'(h) = 0 = 2*a*h + b
-b/(2*a) = h
So we just found the value of h.
To find the value of k, the y-value of the vertex, we need to evaluate the function in the x-value of the vertex, we will get:
k = f(h) = a*( -b/(2*a))^2 + b*( -b/(2*a)) + c
k = b^2/(4a) - b^2/(2a) + c
Then, concluding, we have:
- h = -b/(2*a)
- k = f(h) = b^2/(4a) - b^2/(2a) + c
If you want to learn more, you can read:
https://brainly.com/question/8552341
