Where do the antibodies that trigger the specific immune response come from?
First: What your teacher says about the free floating antibodies is not correct. The first contact with a new antigen to generate specific immunity is done by naive B cells which have a membrane bound B cell receptor and their effector T cells. This contact can either happen directly or by the presentation of the antigen to the cell by antigen presenting cells (APC).

Short answer: 
These naive B cells are present in our body without antigen contact, but are not highly specific.

Long answer:
Our body has the possibility to express a vast variety of different antibodies - estimations go up to 10 billion different antibodies which the human body can produce, each of them capable of binding a different antigen (see reference 1).

To begin with, naive B cells express antibodies as a membrane-bound version, the so called B cell receptor. The antigen specificity of these receptors is random and occurs before the individual had contact with the antigen. These receptors bind a wide variety of different antigens, more or less good. This is the so-called affinity of the receptor (or the antibody) against an antigen.

Upon binding of an antigen, the B cells get activated (if the receive a co-stimulatory signal from a T cell). When activated, the cells starts to proliferate and to differentiate, as shown in the figure below (from here).



Basically, the cells start to re-arrange the genes which are responsible for generation of the variable region of the antibody (the part which does the actual binding of the antigen) to generate highly specific high affinity antibodies to the antigen and then mature into relatively short lived plasma cells, which produce big amounts of antibodies and long lived memory cells. The later are dormant in the body and get reactivated, when the body is challenged by the same antigen, against which it already produced highly specific antibodies.

When re-exposed to an antigen, the antibodies in the blood stream bind to the antigen and mark it for phagocytosis. The phagocytes will then take up the antibody bound antigen, process it and present parts of it on their surface to B/T cells. This either starts the process of selection and clonal expansion as described above or if there are specific memory cells, these cells will proliferate and differentiate into antibody producing plasma cells.

The initial process of adaptive immunity takes about two weeks until highly specific antibodies are available, if you only have to "wake up" the cells from the memory pool, this will take a few days. This is also how vaccinations work.
See also the Wikipedia page on Antibodies for much more details.

References:


Development of the immunoglobulin repertoire.