Answer:
The equation of the function in vertex form is f(x) = -2·(x - 8)² + 6
Step-by-step explanation:
The given values are
x, f(x)
6, -2
7, 4
8, 6
9, 4
10, -2
The equation of the function in vertex form is given as follows;
f(x) = a × (x - h)² + k
To find the values of a, h, and k, we proceed as follows;
When x = 6, f(x) = -2
We have;
-2 = a × (6 - h)² + k = (h²-12·h+36)·a + k.............(1)
When x = 7, f(x) = 4
We have;
4 = a × ( 7- h)² + k = (h²-14·h+49)·a + k...........(2)
When x = 8, f(x) = 6...........(3)
We have;
6 = a × ( 8- h)² + k
When x = 9, f(x) = 4.
We have;
4 = a × ( 9- h)² + k ..........(4)
When x = 10, f(x) = -2...........(5)
We have;
-2 = a × ( 10- h)² + k
Subtract equation (1) from (2)
4-2 = a × ( 7- h)² + k - (a × (6 - h)² + k ) = 13·a - 2·a·h........(6)
Subtract equation (4) from (2)
a × ( 9- h)² + k - a × ( 7- h)² + k
32a -4ah = 0
4h = 32
h = 32/4 = 8
From equation (6) we have;
13·a - 2·a·8 = 6
-3a = 6
a = -2
From equation (1), we have;
-2 = -2 × ( 10- 8)² + k
-2 = -8 + k
k = 6
The equation of the function in vertex form is f(x) = -2·(x - 8)² + 6
