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Quadratic from its values

Solve the first problem 17 problems

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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. Point on the quadrant bisector ●○○○○ 10 ptsPoint $P(a+3,\ 7+a)$ lies on the bisector of the second and fourth quadrants. Find $a$.
3. Values of a quadratic at 0, 1, 2 ●○○○○ 10 ptsFor $p(y)=y^2-y+1$, find $p(0)$, $p(1)$ and $p(2)$.
4. Fitting b through sample means ●●○○○ 20 ptsA sample of six points $(x_1, y_1), \ldots, (x_6, y_6)$ is scattered around the curve $y = bx^2 - \frac{1}{3}$, where $b$ is a real number. Given $\sum_{i=1}^{6} x_i = 11$, $\sum_{i=1}^{6} y_i = 13$ and $\sum_{i=1}^{6} x_i^2 = 21$, find $b$ using the regression line in the variable $t = x^2$. Round to two decimal places.
5. Linear function avoiding the second quadrant ●●○○○ 20 ptsTwo different numbers are chosen from $-2,-1,1,2,3$, the first as $k$ and the second as $b$, to form the linear function $y=kx+b$. Find the probability that the resulting line does not pass through the second quadrant. Give your answer as a decimal.
6. 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.
7. 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 $f(2)$.
8. Finding a from two points on a graph ●●●○○ 50 ptsGiven a constant $a > 0$, the graph of the function $f(x)= \frac{2^{x}}{2^{x}+ax}$ passes through the points $P\left(p, \frac{6}{5}\right)$ and $Q\left(q, -\frac{1}{5}\right)$. If $2^{p+q}=36pq$, find the value of $a$.
9. 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 $a$.

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