Finding the Value of a Quadratic Function: A Solvable Problem for Class 10 Students

Have you ever come across a quadratic function and wondered how to find its value at a specific point? This article walks you through a detailed example of a quadratic function, helping you understand the steps to find its value at a given point, ( f(12) ).

The Given Information and Steps to Solve

We are given a quadratic function ( f(x) ) that attains a minimum value of 2 at ( x 2 ). Additionally, the value of the function at ( x 0 ) is 6. Our goal is to find the value of ( f(12) ).

Step 1: Understanding the Vertex Form

A quadratic function in vertex form is expressed as ( f(x) a(x - h)^2 k ). Here, ( (h, k) ) is the vertex of the parabola. Since the minimum value is 2 at ( x 2 ), we know the vertex is at ( (2, 2) ).

Setting Up the Equation

Substituting ( h 2 ) and ( k 2 ) into the formula, we get:

( f(x) a(x - 2)^2 2 )

We also know that ( f(0) 6 ). Substituting ( x 0 ) and ( f(0) 6 ) into the equation:

( f(0) a(0 - 2)^2 2 6 )

Solving for ( a )

( a(0 - 2)^2 2 6 )

( 4a 2 6 )

( 4a 4 )

( a 1 )

Therefore, the quadratic function is:

( f(x) (x - 2)^2 2 )

Step 2: Finding ( f(12) )

Now, we need to find ( f(12) ):

( f(12) (12 - 2)^2 2 )

( f(12) 10^2 2 )

( f(12) 100 2 )

( f(12) 102 )

Hence, the value of ( f(12) ) is ( boxed{102} ).

Alternative Approach

Writing the General Quadratic Function

Another way to approach this problem is to write the general form of a quadratic function: ( f(x) ax^2 bx c ). We can then convert it into the vertex form.

Given that the minimum value is 2 at ( x 2 ), we can write: