[f4 maths]Functions

1.It is given that(2,5)and(-1,11)a re two points on the graph of the quadratic function y =ax^2+bx+1.
a)find the values of a and b.
b)find the maximum or minimum value of y and the corresponding value of x.
c)write down the vertex and the axis of symmetry of the graph of the function.
a)
a(2)^2 + b(2) + 1 = 5 or 4a + 2b = 4 or 2a + b = 2 ..... (1)
a(-1)^2 + b(-1) + 1 = 11 or a - b = 10 ..... (2)
(1) + (2), 3a = 12 or a = 4
(1)-2(2), 3b = -18 or b = -6
Therefore a = 4, b = -6

b)
4x^2 -6x+1 = y
4(x^2 -3x/2 + 9/16) - 9/4+1 = y
4(x-3/4)^2 -5/4 = y
Therefore, the minimum occurs when x = 3/4 and the minimum is -5/4

c)
The vertex is (3/4. -5/4)
Axis of symmetry is y = 3/4

a) (2,5)and(-1,11)are two points on the graph

5 = a(2)^2 + b(2) + 1
4 = 4a + 2b-------(1)

11 = a(-1)^2 +b(-1) + 1
10 = a - b -----------(2)

a=(20+4)/(4+2)=24/6=4
b=a-10=4-10= -6

b)

y = 4x^2 - 6x + 1
y = 4x^2 - 6x + (6/4)^2 - (6/4)^2 +1
y = (2x -6/4)^2 - (3/2)^2 + 1
y = (2x - 3/2)^2 - 9/4 +1
y = (2x - 3/2)^2 - 5/4

Since (2x - 3/2)^2 has minimum value of 0,
y = (2x - 3/2)^2 - 5/4 has minimum value of -5/4 when x = 3/4

c) The vertex = (3/4, -5/4)

The axis of symmetry is x = 3/4