Above is the graph of the quadratic function (y = ax^{2} + bx + c) where a, b and c are constants. Using the graph, find :
(a)(i) the scales on both axes ; (ii) the equation of the line of symmetry of the curve ; (iii) the roots of the quadratic equation (ax^{2} + bx + c = 0)
(b) Use the coordinates of D, E and G to find the values of the constants a, b and c hence write down the quadratic function illustrated in the graph.
(c) Find the greatest value of y within the range (-3 leq x leq 5).
Explanation
(a)(i) Scale : On x- axis, 2cm = 1 unit
On y- axis, 2cm = 5 units.
(ii) Equation of line of symmetry is x = 1.25.
(iii) Roots of the equation (ax^{2} + bx + c = 0) are x = 0.25 and x = 2.25.
(b) Coordinates are D(0, 1), E(1, -2) and G(3, 4).
Substituting for y and x in (ax^{2} + bx + c = y)
D(0, 1) : (1 = a(0^{2}) + b(0) + c implies c = 1)
E(1, -2) : (-2 = a(1^{2}) + b(1) + c implies -2 = a + b + c)
(a + b = -2 – 1 = -3 … (1))
G(3, 4) : (4 = a(3^{2}) + b(3) + c implies 4 = 9a + 3b + c)
(9a + 3b = 4 – 1 = 3 … (2))
(implies 3a + b = 1 … (2a))
(2a) – (1) : (2a = 4 implies a = 2)
(a + b = -3 implies 2 + b = -3)
(b = -3 – 2 = -5)
(therefore) The equation is (y = 2x^{2} – 5x + 1)
(c) The greatest value of y = 33.5.