Q.246. To find the equation of a quadratic curve with given conditions.

Question 245.
The quadratic function f(x) is negative for x > 9/2 and x < -1, but for no other value of x. If f(1) = 28 determine f(x) algebraically and sketch a graph of the function.

Answer 246.
Let f(x) = ax^2 + bx + c
f(1) = a + b + c = 28 ... ( 1 )
f(-1) = a - b + c = 0 ... ( 2 )
f(9/2) = 81a/4 + 9b/2 + c = 0 ... ( 3 )

Adding eqns. ( 1 ) and ( 2 ),
a + c = 14 ... ( 4 )
Multiplying eqn. ( 1 ) by 9/2 and subtracting from eqn. ( 3 ),
63a/4 - 7c/2 = -126
=> 9a/2 - c = - 36 ... ( 5)
Adding eqns. ( 4) and ( 5 ),
11a/2 = -22 => a = - 4
c = 14 - a = 14 + 4 = 18
b = a + c =18 - 4 = 14
f(x) = -4x^2 + 14x + 18 = - 2(2x - 9)(x + 1)

To sketch the graph of the parabola, f(x)
= - 4 (x^2 - 7x/2 - 9/2)
= - 4 [ (x - 7/4)^2 - 121/16 ]
=> f(x) - 121/4 = - 4 (x - 7/4)^2

This is a parabola with axis parallel to y-axis and vertex at (7/4, 121/4) which is the local maxima. The two arms of the parabola tend to the negative y-direction.

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