Quadratic functions

Maximum value of the quadratic function $f$ is $2$. The graph of $f$ intersects the $x$-axis at the points $[-1;0]$ and $[3;0]$. Find the function $f$.

The graph of the quadratic function $f$ intersects the coordinate axes at the points $[1;0]$, $[4;0]$, $[0;8]$. Find the function $f$.

Suppose, an object, that is in rest, starts to accelerate with the constant acceleration $a$. The distance $s$ travelled by the object in time $t$ is given by the formula $s=\frac{1}{2}a{t}^2$. You can see the graph of the distance $s$ on the time $t$ dependency in the picture. Find the acceleration $a$ of the object.

Consider a simple circuit in which a battery of electromotive force $U_e$ and internal resistance $R_i$ drives a current $I$ through an external resistor of resistance $R$ (see figure). The external resistor could be for example an electric light, an electric heating element, or, maybe, an electric motor. The basic purpose of the circuit is to transfer energy from the battery to the external resistor, where it actually does something useful for us (e.g. lighting a light bulb, or lifting a weight). $$$$ The power $P$ transferred to the external resistor is described by the formula $P=U_{e}I-R_i{I}^2$. What maximum power can be transferred to the external resistor if we have the source with $R_i=0.25\mathrm{\Omega }$ and $U_e=20\mathrm{V}$?