Suppose you are asked to design the first ascent and drop for a new roller coaster By studying the photographs of your favorite coasters, you decide to make the slope of the ascent 0.7 and the slope of the drops -1.5. You decide to connect these two straight stretched $y = L_1(x)$ and $y = L_2(x)$ with part of a parabola $f(x) = ax^2+bx+c$, where $x$ is measured in feet. For the track to be smooth there can't be abrupt changes in direction, so you want the linear segments $L_1$ and $L_2$ to be tangent to the parabola at the points $P$ and $Q$. To simplify the equations, you decide to place the origin at $P (P= (0,0).$ Additionally, suppose the horizontal distance between $P$ and $Q$ is 100ft. 
1.) Write the equations using the variables a,b and c so that the conditions given above are satisfied. Use the equations to determine $f(x) = ax^2+bx+c. $
2.) Determine the coordinates for $Q$. 
3.) Plot $L_1, f$, and $L_2$ together to verify graphically that the roller coaster you are designing is smooth at the points $P$ and $Q$. 
I tried finding the derivative of $ax^2+bx+c$ and used that, $2ax+b$, to find the value of $b$ which is 0.7. I also found the equation of line one which is $y=0.7x$ at $p (0,0)$ and it is tangent. I also determined that $a = -0.011$ from the derivative. So Im missing the $c$ value for $-.011x^2+0.7x+c=y$ and the coordinates for point $Q$ along with its equation $L_2$.  

1.) $f(x) = -0.011x^2+0.7x$
2.) $Q:(100,-40)$
3.) $L_1 = y = .7x$ $L_2 = (y+40)=-1.5(x-100)$
