Given is the graph of the relation (y = ax^{2} + bx + c) where a, b and c are constants. Use the graph to :
(a) find the roots of the equation (ax^{2} + bx + c = 0);
(b) determine the values of constants a, b and c in the relation using the values of the coordinates P and Q and hence write down the relation illustrated in the graph
(c) find the maximum value of y and the corresponding value of x at this point.
(d) find the values of x when y = 2.
Explanation
(a) The roots of the equation are -1.5 or 2.
(b) C = 6 (intercept on the positive vertical axis)
(ax^{2} + bx + c = 0)
When x = 2, c = 6
(4a + 2b + 6 = 0 implies 4a + 2b = -6 ….(1))
When x = -1.5, c = 6
(a(-1.5)^{2} + b(-1.5) = -6)
(frac{9a}{4} – frac{3b}{2} = -6 implies 9a – 6b = -24 … (2))
Multiply (1) by 3 : (12a + 6b = -18 … (1a))
(1a) + (2) : (12a + 9a = -18 – 24 implies 21a = -42)
(a = -2)
(4a + 2b = -6 implies 4(-2) + 2b = -6)
(2b = -6 + 8 = 2 implies b = 1)
(a, b, c) = (-2, 1, 6)
Hence the relation illustrated in the graph is (y = -2x^{2} + x + 6).
(c) When (x = 0.2, y_{max} = 6.2)
(d) When (y = 2, x = text{ -1.2 or 1.7})