Great question! If I understand your situation correctly, you are using a team site as the top level site for your site collection. Therefore by default, the advanced search does not appear to the right of the search box.
This is expected behaviour as the advanced search function requires the existence of a page to perform the advanced search which typically comes by default with a collaboration portal template via the search center subsite that is included. You can create custom search pages but the easiest way for you to accomplish what you want is to follow steps below.
To activate advanced search on your team site, I recommend the following simple steps:
1. Create a subsite using the Search Center template (under Enterprise tab in template selection window). When the site is created, click the Advanced Search link and copy the url of the page you are redirected to.
2. From the Site Settings page of your team site (the top level site), select search settings and select the radio button that appears next to phrase "Use custom scopes. Display richer results using the following Search Center:"
3. Enter the url of the advanced search page from the site you created in step 1 and Click OK.
Once you do this, you should see the Advanced Search appear to the right of your search box on your team site.
Hope this helps!
MVP - Microsoft Office SharePoint Server