It’s incorrect to say that there are only three auxiliary verbs in English, namely “do,” “be,” and “have” as you’ve listed. The count actually rises to as many as 23 when we include the so-called modal auxiliaries; however, “do,” “be,” and “have” hold the distinction of being the three primary auxiliary verbs in English in the sense that they are the most commonly used.
Your listing of the different forms of the auxiliary verb “do”—“does,” “do,” and “did”—is complete. In the case of “have,” however, the progressive-tense form “having” has to be added to make a total of four: “has,” “have,” “had,” and “having.” And for “be,” you listed only its three present-tense forms “am,” “is,” and “are”; to these must be added the past tense forms “was” and “were,” the progressive tense form “being,” and the past-participle “been,” making a total of seven forms.
Before making a complete accounting of the English auxiliary verbs, let’s agree first on a definition of the term “auxiliary verb.” A simple but instructive definition is that an auxiliary verb—also loosely called a “helping verb” or more precisely as “verbal auxiliary”—is one that enables or helps a main verb express tense, voice, emphasis, or modality. Another way of saying this is that an auxiliary verb adds functional or grammatical content to the information expressed by the main verb.
As examples, in the sentence “They did take the loot,” the verb “did” works as an intensifier for the verb “take” to emphasize that the action was, in fact, done; in the sentence “He is being fooled,” the auxiliary verb “being” works with the linking verb “is” to form the present progressive passive tense of the main verb “fooled”; and in the sentence “She has taken my share of the cake,” the auxiliary verb “has” works with the past participle “taken” to form the present perfect tense of the verb “take.”
There’s a subclass of auxiliary verbs that, unlike the three primary auxiliary verbs you listed, don’t inflect or can’t change form at all. These are the so-called modal auxiliaries, or modals for short. The most commonly used modals are “can,” “could,” “may,” “might,” “must,” “shall,” “should,” “will,” and “would”; less commonly used are the modals “dare,” “need,” and “ought.”
Functionally, a modal auxiliary or modal works with a main verb to express conditionality, necessity, obligation, ability, or wishful desire; it is unlike the typical auxiliary verb, which works with the main verb to denote voice, tense, or emphasis. In the sentence “She can speak French fluently,” for instance, the modal “can” works to convey the ability of the subject to speak French fluently. In the sentence “She does speak French fluently,” in contrast, the auxiliary verb “do” acts as an intensifier to emphasize the subject’s ability to, in fact, speak French fluently.
Keep in mind, though, that the three auxiliary verbs “be,” “do,” and “have” can also function as main verbs. For instance, in the sentence “You be the one,” the verb “be” works as a main verb to denote asking someone to assume a certain role; in “She does all the work here,” the verb “does” functions as the main verb to denote performing all the work; and in “They have lots of money,” the verb “have” works as a main verb to denote possession of lots of money.
