The graph of the function f(x) = ax2 + bx + c has vertex at (1, 4) and passes through the point (-1, -8). The coefficient a is
the vertex tells us that
f(x) = a(x-1)^2 + 4
using the point (-1,-8) we see that
a(-1-1)^2 + 4 = -8
4a+4 = -8
a = -3
f(x) = -3(x-1)^2 + 4 = -3x^2 + 6x + 1
Vertex has coordinate x = - b / 2a
In this case:
x = 1
x = - b / 2a
1 = - b / 2a
Multiply both sides by 2a
2 a = - b
Multipy both sides by - 1
- 2 a = b
b = - 2 a
Put this value in equation:
f(x) = a x² + b x + c
with coordinates:
x = 1
f = 4
4 = a • 1² + ( - 2 a ) • 1 + c
a - 2 a + c = 4
- a + c = 4
Add a to both sides.
c = a + 4
Again quadratic equation with coordinates:
x = - 1 , f = - 8
x² + b x + c = f(x)
a ( - 1 )² + ( - 2 a ) • ( - 1 ) + a + 4 = - 8
a + 2 a + a + 4 = - 8
4 a + 4 = - 8
Subtract 4 to both sides.
4 a = - 12
a = - 12 / 4
a = - 3
By the way:
b = - 2 a = - 2 • ( - 3 ) = 6
c = a + 4 = - 3 + 4 = 1
Your quadratic equation is:
f(x) = - 3 x² + 6 x + 1