A random variable $Y \sim U[0,1]$. Let $W = \frac7{10} - b\cdot(\frac7{10} - y)^2, 0<b<1$.
Completely specify the CDF and PDF of $W$.
Also show that the PDF of W integrates to $1$.
So I have worked through most of this problem. Importantly, the transformation is $2:1$ on the interval y $\in (\frac25,1]$ and $1:1$ on the interval y $\in [0, \frac25)$. Thus I think the CDF is:
$$\begin{cases} \frac3{10}+\sqrt{\frac{7/10 - w}b } & \text{for}~~ w \in (\frac7{10} - \frac{49b}{100} ,\, \frac7{10} - \frac{9b}{100}) \\ 2\sqrt{ \frac{7/10 - w}b } & \text{for}~~ w \in (\frac7{10} - \frac{9b}{100} ,\, \frac7{10}) \\ \end{cases}$$
Can anyone confirm this? If I know this is right, I can figure out the rest on my own. When integrating the PDF, I am running into some strange (${}+{}$ or ${}-{}$) situations that integrate to $1$ only if I select a certain sign, so I am having trouble verifying the CDF is correct by integrating the PDF over its support.