Leading coefficient
SAT glossary · Math
Definition
The coefficient a on the highest-power term. It controls direction (a>0 opens up, vertex is the minimum; a<0 opens down, vertex is the maximum) and width (large |a| narrow, small |a| wide). When completing the square, whatever is subtracted inside the parentheses emerges multiplied by it — the error the vertex-form item is built on.
Leading coefficient in plain English
The leading coefficient is just the number multiplying the highest-power term in a polynomial — in ax^2+bx+c, that's the letter a. For a parabola, this single number controls two things at once: which way it opens (if a is positive, it opens upward like a smile and the vertex is the LOWEST point; if a is negative, it opens downward like a frown and the vertex is the HIGHEST point), and how wide or narrow it looks (a bigger |a| makes it skinnier, a smaller |a| makes it wider). It also secretly reappears during 'completing the square': whatever number you subtract inside the parentheses comes back out MULTIPLIED by a — forgetting that multiplication is exactly the mistake that vertex-form questions are designed to catch.
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Practice Leading coefficient
The function f is defined by f(x)=3x^2-12x+7. What is the minimum value of f(x)?
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