Given $ f(1) = c(1)^2 = 3 \Rightarrow c = 3 $, so $ f(x) = 3x^2 $.

Thu, 16 Apr 2026 07:49:42 +0000

Understanding the Quadratic Function: $ f(x) = 3x^2 $ Explained Mathematics thrives on patterns and clarity—and few functions exemplify a clear quadratic relationship as effectively as $ f(x) = 3x^2 $. But how do we arrive at this precise formula from a single data point? In this article, we’ll explore the derivation, significance, and applications of the specific function $ f(x) = 3x^2 $, starting with the foundational step: $ f(1) = c(1)^2 = 3 $, which leads directly to the value $ c = 3 $, defining the function completely. ### From a Given Point to a Defined Function The equation $ f(1) = c(1)^2 $ seems straightforward but holds deep mathematical importance. Substituting $ x = 1 $ into the general quadratic form $ f(x) = ax^2 + bx + c $, we get:
$$
f(1) = a(1)^2 + b(1) + c = a + b + c
$$
But in this case, the function is simplified as $ f(x) = c x^2 $, implying a zero coefficient for $ x $ and constant $ a = 0 $, $ b = 0 $. This means $ f(x) = c x^2 $ is fully determined by the value of $ c $—specifically, $ f(1) = c $, which is given as $ 3 $.