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.