Answer:
f(x) = (x - 2)² + 2
Step-by-step explanation:
The vertex form of the quadratic function is:
f(x) = a(x - h)² + k
where:
(h, k) = vertex
The axis of symmetry is the imaginary vertical line where x = h
a = determines whether the graph opens up or down, and how wide or narrow the graph will be.
h = determines the horizontal translation of the parabola.
k = determines the vertical translation of the graph.
Given the vertex occurring at point (2, 2), along with one of the points on the graph, (4, 6):
Substitute these values into the vertex form of the quadratic function:
f(x) = a(x - h)² + k
6 = a(4 - 2)² + 2
6 = a(2)² + 2
6 = 4a + 2
Subtract 2 from both sides:
6 - 2 = 4a + 2 - 2
4 = 4a
Divide both sides by 4:
4/4 = 4a/4
1 = a
Therefore, the quadratic function in vertex form is: f(x) = (x - 2)² + 2
