Find Coefficients in y = 6x - 6x² + 3: Identifying a, b, and c

Let's recall the general form of a quadratic function:

$y=ax^2+bx+c$

Examine the given function in the problem:

$y=6x-6x^2+3$$$

Note that in the general form of the quadratic function mentioned above, the terms are arranged from the highest power (which is the quadratic term - power of 2) to the lowest power (which is the free term - power of 0),

Therefore, in order to make it easier to identify the coefficients, we'll apply the commutative property of addition and rearrange the terms of the quadratic function so they are written from highest to lowest power:

$y=6x-6x^2+3 \\ y=-6x^2+6x+3$

We can then identify that the coefficient of the quadratic term, meaning the coefficient of the term with power two: $a$ is $-6$ We'll continue and identify that the coefficient of the term with power one: $b$ is $6$ and finally we'll identify that the coefficient of the term with power 0, meaning the free term: $c$ is $3$

To summarize, the coefficients in the given function are:

$a=-6,\hspace{4pt}b=6,\hspace{4pt}c=3$

Therefore, the correct answer is answer A.

Note:

The coefficient $c$is the free term - and we said before that it's the coefficient of the term with power zero - $x^0$this is because any number different from zero raised to the power of zero equals 1:

$x^0=1$, and therefore we could write the general form of the function above as:

$y=ax^2+bx+c \\ \downarrow\\ y=ax^2+bx+c\cdot1 \\ \downarrow\\ y=ax^2+bx^1+c x^0$

meaning, $c$ is the coefficient of the term with power 0.