Correlation($U,V$)=Correlation($X,Y$) Let $X$ and $Y$ be random variables such that $0<\sigma^2_X<\infty$ and $0<\sigma^2_Y<\infty$. Suppose that $U=aX+b$ and $V=cY+d$, where $a \not= 0$ and $c \not= 0$. Show that $\rho(U,V)=\rho(X,Y)$ if $ac>0$, and $\rho(U,V)=-\rho(X,Y)$ if $ac<0$.

Might be best to start with covariance and $a,c > 0$.
$$Cov(U,V) = Cov(aX+b, cY+d) = Cov(aX,cY) + Cov(aX,d) + Cov(b,cY) + Cov(b,d)\\
= Cov(aX,cY) = acCov(X,Y).$$
Then $\rho(U,V) = Cor(U,V) = \frac{Cov(U,V)}{SD(U)SD(V)}.$
Finally, finish by finding $\sigma_U = SD(U)$ and $\sigma_V = SD(V).$
I hope my notation is sufficiently similar to notation in your text so that
you can follow this.
