If you've read The Tempest (and you're not just skimming this post as a replacement for actual reading the night before your exam), I'm willing to bet that you were not paying attention to "family relationships" as a theme. The other themes in the play, sure- like power and magic- but family relationships just seems like a theme considerably more extraneous than the others to be analysed in a play with so many things going on. That said, the school system is often paradoxical in its pointlessness, so let's get down to analysin'.
Family Relationships are those interpersonal bonds that exist by virtue of like descent or familial connections through other means. When we speak of the theme of family relationships, we most often refer to the specifics and details of how the people involved in the familial relationship interact with each other and the implications of their relationship.
There exist two main types of family relationships in the play, parent-child relationships and sibling relationships (the only one of major significance being that between Prospero and Antonio).
Parent-Child Relationship
Miranda and Prospero
The relationship between these two characters is presented with the tenderness and concern typical of a father-daughter relationship. They show mutual concern and affection, with Prospero's pride in his daughter coming out through dialogue numerous times in the play.
Miranda evidently holds endearment for her father, referring to him as 'my dearest father,' while also acknowledging the strength and potency of his art, seen in how she pleads with him in the beginning of act 1 scene 2 to ally the "wild waters" that caused the wreckage of the ship at sea.
Prospero values his daughter very highly, not only because of her identity as his sole heir, but also due to her centrality to his plan. Most of his interactions with his daughter communicate the care and concern he holds for her, such as "I have done nothing but in care of thee, of thee, my dear one, thee my daughter." Lines such as this seem to give the impression that he wants her to see him in a positive light, considering that one line prior he reassures her "there's no harm done," painting his art in a benign light. Caliban remarks about Prospero's value of his daughter- 'he himself calls her a nonpareil.' The magus is also very voluble with his praise of Miranda, assuring Ferdinand when he blesses their union that "she will oustrip all praise and make it halt behind her."
Prospero's role as father in their relationship also results in his making unilateral decisions for her protection, most notably his choice to put Ferdinand through trials of strenuous labour before he can be allowed to wed Miranda. This is, of course, to ensure the depth of the prince's love, as Prospero could not fathom handing off "that for which [he] lives" to a shallow monster.
It should be noted that, while Miranda is deferential towards Prospero, she is not beyond defying him or protesting his decisions. In the end of Act 1 Scene 2 when she encounters Ferdinand for the first time, she pleads with her father that the man she sees is "gentle and not fearful," convinced that "there's nothing ill that can dwell in such a temple." She opposes him three times despite his denial of her pleas, hanging on his garments in protest. Later on, in act 3 scene 1, she disobeys a direct command by her father not to tell Ferdinand her name in the middle of a visitation with the man unwarranted by Prospero. Thus, while respectful of her father, she is not completely submissive to his wishes, and Prospero, while prizing his daughter dearly, is not willing to let her have her every whim and fancy- especially in her first romance.
Ferdinand and Alonso
The king and his son are parted in Act 1 Scene 1, and only reunite in Act 5 Scene 1 when Prospero reveals the son playing chess with Miranda. Throughout the play, however, the dialogue of either character is littered with expressions of grief and pain out of the loss of the other.
Alonso's first lines after the shipwreck are reflective of the deeply distraught emotion he feels out of the loss of his son and sole heir:
"O thou mine heir of Naples and Milan, what strange fish hath made his meal on thee?"
He obviously values his son, and even desires to die with his son out of the guilt he feels for supposedly causing his death as recompensation for the perfidy he committed against Prospero:
"Therefore my son i'th' ooze is bedded, and I'll seek him deeper than e'er plummet sounded and there with him lie mudded."
Ferdinand seems similarly distraught by the loss of his father:
"[My father] does hear me, and that he does, I weep. Myself am Naples, who, with mine eyes, never since at ebb, beheld my father wrecked."
Sibling Relationship
Prospero and Antonio
The most interesting aspect of the relationship between these two Dukes of Milan is the value each one assigns the other. Because of the treachery committed by Antonio 12 years prior, the relationship between the two is cold, bitter and completely devoid of the fraternal warmth one would expect in a typical sibling relationship. Being reunited with his brother after 12 years does not warrant even a single word from Antonio in the fifth act, and likewise, Prospero feels unable to consider the Svengali before him a brother: "most wicked sir, whom to call brother would even infect my mouth."
Prospero's initial attitude to his brother (before the betrayal) was one of great love, as he tells Miranda, "whom, next thyself, of all the world I loved." He had trusted him greatly, with "a confidence sand bound," until he plots to commit fratricide to further his own power and influence. Though Prospero forgives the "unnatural" brother by the end of the play, his agitation and anger with that treachery is seen throughout the prior acts.
Antonio seems untroubled by what he did to his own brother, claiming to have no conscience: "Ay sir, where lies that [conscience]? I feel not this deity in my bosom." Strangely, the murder of his sibling seems to be a crime for which he does not repent, remaining basically completely silent after Act 3 Scene 3 when Ariel condemns him, Alonso and Sebastian for their involvement in the treachery. He even expresses pride in what he has done, flaunting the ducal robes gained from the sacrifice of his brother to Sebastian to convince him to do the same in pursuit of the throne:
"I remember you did supplant your brother Prospero."
"True: and look how well my garments sit upon me much feater than before. My brother's servants were then my fellows, now they are my men."
Prospero seems, to him, a step along the way to his ascension into power- and it is this problematic Machiavellian nature of Antonio's that creates the coldness in this relationship.
Dramatic Devices Presenting Family Relationships
Monologue
Prospero's extended flashback in Act 1 Scene 2 presents the nature of Antonio's betrayal, making the premise for the frigid state of their relationship clear.
I pray thee, mark me.
I thus neglecting worldly ends, all dedicated
To closeness and the bettering of my mind 90
With that which, but by being so retired,
O'er-prized all popular rate, in my false brother
Awaked an evil nature, and my trust,
Like a good parent, did beget of him
A falsehood in its contrary as great
As my trust was, which had indeed no limit,
A confidence sans bound...
To have no screen between this part he played
And him he played it for, he needs will be
Absolute Milan. Me, poor man, my library
Was dukedom large enough. Of temporal royalties
He thinks me now incapable; confederates,
So dry he was for sway, wi'th' King of Naples
To give him annual tribute, do him homage,
Subject his coronet to his crown, and bend
The dukedom yet unbowed (alas, poor Milan)
To most ignoble stooping...
Mark his condition and th'event, then tell me
If this might be a brother.
Now the condition.
This King of Naples, being an enemy
To me inveterate, hearkens my brother's suit,
Which was that he, in lieu o'th' premises
Of homage, and I know not how much tribute,
Should presently extirpate me and mine
Out of the dukedom and confer fair Milan,
With all the honours, on my brother.
Dialogue
Certain things said in dialogue describe the nature of these character's relationships:
"most wicked sir, whom to call brother would even infect my mouth."
"Therefore my son i'th' ooze is bedded, and I'll seek him deeper than e'er plummet sounded and there with him lie mudded."
"[My father] does hear me, and that he does, I weep. Myself am Naples, who, with mine eyes, never since at ebb, beheld my father wrecked."
"O thou mine heir of Naples and Milan, what strange fish hath made his meal on thee?"