Parametrizing a given line and equations

Question

1) Parametrizethe given line contraining the points (3,2) and (-5,6).

2) Find the parametric equations for the segment joining the given points (2,3) and (5,5) where $0\leq t \leq 1$.

3) Find the parametric equation for the segment joining the given points (-3,0) and (1,6) where $0\leq t \leq 2$.

Im just learning parametric curves and these questions have me on holdup. My attempt is:
1) $c(t) = ( 3-8t, 2+2t )$
2) $c(t) = ( 2+5t, 1+2t )$

Im not sure what im doing wrong but both of the answers are not correct. For the third one, I dont know what to do.

Answer 1

Just miscalculated/miscopied:

1. $c(t)=(3-8t,\,2+4t)$
2. $c(t)=(2+3t,\,3+2t)$
3. $c(t)=(-3+\frac42t,\,0+\frac62t)$.

Answer 2

Hint: Is there a pair of values of $t$ in each of your parametrisations that give the desired result?

For instance, your first attempt at the first question. Obviously, when $t=0$, it matches the first point. But when $t=1$, you get $(-5,4)$, and not $(-5,6)$.

Hint 2: The slope has to match between the two points. That is, "rise over run". So if your parametrisation is $(at+b,ct+d)$, then rise over run is $\frac{ct_1-ct_0}{at_1-at_0}=\frac{c}a$. This must match the slope of a line between the two points, so for the first question, it's $\frac{6-2}{-5-3}=\frac4{-8}=-\frac12$.