CAT Online Coaching – A Good one from Quadratic Equations

Question

A quadratic function f(x) attains a maximum of 5 at x = 2. The value of function at x=0 is 1. Find the quadratic expression.

Solution

Any quadratic equation can be written in two ways, 1) C – a(x-k)² and 2) a(x-k)² – C.

These ways of writing the quadratic equation simply make use the fact that anything that is squared is always positive. So, for a minimum value of the function, we can express the quadratic function as sum of Squared Term + Minimum Value and for the Maximum value of the function, we can express it as Maximum Value – Squared Term.

So for this question, f(x) can be written as –> C – a(x-k)².

Now, it is given that maximum value of f(x) is attained at x = 2. => C=5; since we will get the maximum value only if it is subtracted by the least quantity, k = 2.

5 – a(x-2)². It is given that at x = 0, value of the function is 1. This means,

5 – a(-2)² = 1 -> 5 – 4a = 1 -> a = 1.

So the expression is 5 – (x-2)². So the quadratic expression is 5 – (x² – 4x + 4) => -x² + 4x + 1.