Here is one solution:
It's not a very good looking curve, but, given the strange shape of your point set, I'm not sure there are solutions that are much better.
The black dots are the original data points, and the alternating red and green pieces are the quadratic segments.
As you can see, the "break points" or "knot points" where segments join are not at the original data points. This is fairly typical of what you see when interpolating using splines with even degree.
The curve was calculated using pretty standard spline interpolation methods -- essentially you just write down a set of linear equations that express the interpolation conditions, and solve them. The details are left to the student :-).
The curve is C1.
My guess is that this is not what your instructor was expecting you to do. He was expecting you to somehow make up some tangent vectors at the data points, and use the points and tangents to construct the spline. If you do this, you will run into trouble wherever there's an inflexion. Maybe that was the point of the exercise.
If you want to do it this way, I would recommend that you proceed as follows:
(1) Make up the tangent vectors. There are many ways to do this.
(2) Take each pair of points in turn.
(a) If its possible to build a quadratic segment from the two points and two tangents, then insert a quadratic.
(b) If a quadratic is not possible (because of an inflexion, or a 180 degree turn), build a Bezier cubic from the two points and two tangents, instead. Then split this cubic into two, and then replace each of the two halves with a quadratic. You should split at the inflexion point, if there is one.
Here's an example of what you can get by this method. The pink points are places where I joined together two quadratics to replace a cubic, as mentioned in 2(b) above:
I know this is all a bit vague and sketchy, but it should give you some hints, at least.