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7 problems match “quadratic functions”. Clear

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1. Weighted sum of quadratic values ●○○○○ 10 ptsConsider the function $f(x) = 2x^2 - 4x + 9$. Evaluate $2f(3) + 3f(-3)$.
2. Coefficients of a re-written quadratic ●○○○○ 10 ptsThe expression $2(x-3)^{2}-12$ can be re-written as $ax^{2}+bx+c$ for some numbers $a, b, c$. What is the value of $10a-b-4c$?
3. Minimum of f(m) + f'(n) on an interval ●●○○○ 20 ptsThe function $f(x) = -x^3 + ax^2 - 4$ has a local extremum at $x = 2$. For $m, n \in [-1,1]$, find the minimum value of $f(m) + f'(n)$, where $f'$ is the derivative of $f$.
4. Corrected value in a quadratic table ●●○○○ 20 ptsA student plots $y=ax^2+bx+c$ ($a\neq 0$) using seven equally spaced $x$-values $x_1<x_2<\cdots<x_7$, where $x_{k+1}-x_k$ is constant. The recorded $y$-values are $51, 107, 185, 285, 407, 549, 717$ for $x_1,\ldots,x_7$ respectively. Exactly one of these $y$-values was calculated incorrectly. Find the correct value of that $y$-value.
5. Value of f(2) for a quadratic ●●●○○ 50 ptsLet $a$ and $b$ be real numbers. The quadratic function $f(x)=x^{2}+ax+b$ satisfies $f(f(0))=f(f(1))=0$ and $f(0)\neq f(1)$. Find $f(2)$.
6. Smallest bound on derivative at zero ●●●○○ 50 ptsFind the smallest real number $A$ such that for every quadratic polynomial $f(x)$ satisfying $|f(x)| \leqslant 1$ for all $0 \leqslant x \leqslant 1$, the inequality $f'(0) \leqslant A$ holds.
7. Coefficient of a quadratic from its values ●●●●○ 100 ptsThe function $f(x)$ is of the form $ax^{2}+bx+c$ for some integers $a$, $b$ and $c$. Given that $\{f(177883), f(348710), f(796921), f(858522)\} = \{1324754875645, 1782225466694, 1984194627862, 4388794883485\}$, compute $a$.