Determine the quadratic function of f whose vertex is (2,-1) and passes through (3,1)
f(x)=
from the vertex, we know that
y = a(x-2)^2 - 1
since y(3) = 1,
1 = a(3-2)^2-1
1 = a-1
a = 2
y = 2(x-2)^2 - 1
To find the quadratic function f(x) that satisfies the given conditions, we can use the standard form of a quadratic function:
f(x) = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
Given that the vertex is (2, -1), we can substitute these values into the equation:
f(x) = a(x - 2)^2 - 1
To determine the value of 'a', we need to use the second point, (3, 1), that lies on the parabola. Substituting these coordinates into the equation, we get:
1 = a(3 - 2)^2 - 1
1 = a(1)^2 - 1
1 = a - 1
a = 1 + 1
a = 2
Now that we have the value of 'a', we can substitute it back into the equation:
f(x) = 2(x - 2)^2 - 1
Therefore, the quadratic function f(x) that satisfies the given conditions is:
f(x) = 2(x - 2)^2 - 1