Question Type 3: Formulating and solving linear equations in coefficients of polynomials Exercises

Determine the quadratic $f(x)=ax^2+bx+c$ with $c=3$ passing through $(1,6)$ and $(2,11)$.

Determine the equation of the line through $(1,2)$ and $(3,6)$.

The graph of the function $q(x)=ax^2+bx+1$ passes through the points $(1,4)$ and $(2,9)$. Find the values of $a$ and $b$.

Given $s(x)=ax^2+bx-1$ and $s(1)=2$, $s(2)=9$, and $s(3)=20$, find the values of $a$ and $b$.

Find the quadratic function $p(x) = ax^2 + bx + c$ that passes through the points $(1, 0)$, $(2, 3)$, and $(3, 8)$.

Find the equation of the line passing through the points $(4,0)$ and $(3,1)$.

Determine the quadratic function $f(x)=ax^2+bx+c$ passing through $(0,1)$, $(1,4)$ and $(2,9)$.

Find the quadratic $f(x) = ax^2 + bx + c$ satisfying $f(1) = 3$, $f(2) = 8$, and $f(-1) = 5$.

The function $r$ is defined by $r(x)=2x^2+bx+c$, where $b, c \in \mathbb{R}$.

Given that $r(1)=5$, $r(2)=12$, and $r(3)=23$, find the expression for $r(x)$.

Find the function of the form $h(x)=ax^2+bx+c$ passing through $(0,-2)$, $(1,-1)$ and $(3,7)$.

Find the equation of the line passing through $(-2,5)$ and $(4,-1)$. Give your answer in the form $y = mx + c$.