Path homotopy between $\alpha(t) = t-t^2$ and $\beta(t) = t^2-t$

Question

I am trying to find a path homotopy between $\alpha(t) = t-t^2$ and $\beta(t) = t^2-t$ where $t\in[0,1]$

$\alpha$ and $\beta$ are path homo topic if they have the same endpoints, $p, q$ and $\exists F: I \times I \rightarrow X$ such that $F(s, 0)=\alpha(s)$, $F(s, 1)=\beta(s)$, $F(0, t)=p$, $F(1, t)=q$

Clearly $\alpha(t)=-\beta(t)$

Also, $\alpha(0)=\beta(1)$ and $\alpha(1)=\beta(0)$ so they have the same endpoints.

I have no idea how to find a function $F$ as defined above, i.e. a path homotopy. Could you help me? Thanks

Answer 1

Find a continuous function $r(t)$ so that $r(0)=1$ and $r(1)=-1$. Then let $F(s,t)=r(t)\alpha(s)$. A linear function $r$ will do just fine.