Maxima and minima
· Optimal value means either maximum or minimum point of the parabola.
· We use method of completing square to find the maximum or minimum value from (y = ax^2 + bx + c) standard form of the equation.
· We use method of completing square to find the maximum or minimum value from (y = ax^2 + bx + c) standard form of the equation.
Example: Simple Trinomial x^2 + bx + c = y
Find the minima or maxima of y = x^2 + 8x + 5. y = x^2 + 8x + 5 y = x^2 + 8x + a – a + 5 y = x^2 + 8x + 16 -16 + 5 y = (x^2 + 8x + 16) – 16 + 5 y = (x + 4)^2 – 11 Therefore the vertex is (-4, -11), and minima is -11, as the following equation represents a positive parabola. |
Example: Complex Trinomial ax^2 + bx + c = y
y = 2x^2 + 12x + 11 y = 2(x^2 + 6x) + 11 y = 2(x^2 + 6x + a – a) + 11 y = 2(x^2 + 6x + 9 – 9) + 11 y = 2(x^2 + 6x + 9) – 18 + 11 y = 2(x + 3)^2 + 7 Therefore the vertex is (-3, 7), and the minima is 7, as the following equation represents a positive parabola. |