Step 1 :The function given is \(f(x) = 10 - 3x^2\).
Step 2 :The derivative of a constant is 0 and the derivative of \(x^2\) is \(2x\), so the derivative of the function is \(f'(x) = -6x\).
Step 3 :Substitute \(a=2\) into the function and its derivative: \(f(2) = 10 - 3(2)^2 = 10 - 12 = -2\) and \(f'(2) = -6(2) = -12\).
Step 4 :The equation of the tangent line is: \(L(x) = -2 - 12(x - 2)\).
Step 5 :Simplify this equation to get: \(L(x) = -2 - 12x + 24 = 22 - 12x\).
Step 6 :Final Answer: The linear approximation of the function at \(a=2\) is \(L(x) = \boxed{22 - 12x}\).