The following shows the graph of $y=a x^{2}+b x+c$.
G10.1 Find $c$.
G10.2 Find $a$.
G10.3 Find $b$.
G10.4 If $y=x+d$ is tangent to $y=a x^{2}+b x+c$, find $d$.
$\begin{array}{l}x=0, y=c=3 \\ y=a\left(x+\frac{1}{2}\right)(x-3) \\ \text { Sub. } x=0, y=3 \\ \Rightarrow-\frac{3}{2} a=3 \\ a=-2 \\ 3-\frac{1}{2}=\text { sum of roots }=-\frac{b}{(-2)} \\ b=5 \\ \text { Sub. } y=x+d \text { into } y=a x^{2}+b x+c \\ -2 x^{2}+5 x+3=x+d \\ 2 x^{2}-4 x+d-3=0 \\ \Delta=(-4)^{2}-4(2)(d-3)=0 \\ 4-2 d+6=0 \\ d=5\end{array}$