A metaphor for people living up or down to their reputation

I have been asked to improve my question.

Can anyone help with the full quote and derivation thereof for a metaphor that begins “give a dog a bad name and hear (or see) him bark”?

My understanding is that it involves someone living up or down to their reputation. “They told her she would fail and she failed. Give a dog a bad name and see him bark”.

Can anyone help please? Thanks in advance.

"Treat a man as he is, and he will remain as he is. Treat a man as he could be, and he will become what he should be.” - Ralph Waldo Emerson

Perhaps this? Though I suspect you are looking for something shorter

One dog-centric proverb that may be on point here is "Give a dog a bad name and hang him." Here is the entry for that saying in Martin Manser, The Facts on File Dictionary of Proverbs (2002):

give a dog a bad name and hang him Once somebody's reputation has been damaged—for example, by rumor or slander‚it will never recover: "The Liberal impulse is almost always to give a dog a bad name and hang him: to denounce the menaced proprietors as enemies of mankind, and ruin them in a transport of virtuous indignation" (George Bernard Shaw, The Intelligent Woman's Guide to Socialism, 1928). The proverb was first recorded, with different wording,in 1706.

Variant of this proverb: he that has an ill name is half hanged.

Proverb expressing similar meaning: throw dirt enough and some will stick.

I don't know what wording Manser found in his 1706 source, but the following form appears in James Kelly, A Complete Collection of Scottish Proverbs: Explained and Made Intelligible to the English (1721), with the attendant contextual note:

Give a Dog an ill Name, and he'll soon be hang'd.

Spoken of those who raise an ill Name on a Man on purpose to prevent his Advancement : A cursed, but common, Practice.

John Ray, A Collection of English Proverbs (1732) has this related expression (and comment):

He that would hang his dog, gives out first that he's mad.

He that is about to do any thing disingenuous, unworthy, or of evil fame, first bethinks himself of some plausible pretence.

This saying is quite old, appearing in John Clarke, Paroemiologia Anglo-Latina in Usum Scholarum Concinnata. Or Proverbs English, and Latine, Methodically Disposed According to the Common-place Heads, in Erasmus His Adages (1639):

He that would hang his dog, gives out first that he is mad.

And in John Ley, Light for Smoke: or, A Cleare and Distinct Reply: by Iohn Ley, One of the Assembly of Divines at Westminster, to a Darke and Confused Answer in a Booke Made, and Intituled The Smoke in the Temple (1646):

It is true, that according to the French Proverbe, He that would have his neighbours dog hang'd gives out that he is mad; so your party that would have the Presbyteriall Government literally suspended, and made away as a mad dog, ...

Similarly, from "A Declaration of Sir John Gaire Lord Mayor· Alderman Langham Alderman Adams Alderman Bunce Sheriff Cullam" (1647):

But (though a homely, yet) it is a true saying; He that is willing to hang his neighbours dog, must say he steals sheep.

Interestingly, a statute in passed in the Massachusetts colony in 1658 specifically called for hanging a dog that has killed one or more sheep. From The Generall Laws of the Massachusets Colony, Revised and Published, by Order of the General Court in October 1658 (1658):

It is further Ordered; That if any man shall course Sheep with a Dog, or otherwise molest them, by driving them from their feeding, he shall pay five shillings for every such offence, besides double damages, and if any dog shall kill any sheep, the Owner shall either hang such dog, or pay double damages for the sheep, and if any dog hath been seen to course or bite Sheep before, not being set on, and his Owner hath had notice thereof, then he shall both hang his dog, and pay for such Sheep as he shall either bite or kill; And if in such case he shall refuse to hang his dog, then the Constable of the Town upon notice thereof, shall forthwith cause it to be done.

Another unusual antecedent of the "give a dog a bad name and hang him" proverb is this joke recorded in Humphrey Crouch, England's Jests Refin'd and Improv'd Being a Choice Collection of the Merriest Jests, Smartest Repartee's, Wittiest Sayings, and Most Notable Bulls, yet Extant with Many New Ones, Never Before Printed (1687):

One perswaded a man to hang his Dog that had done some mischief, I am loath to hang him, says he, but I'll go amongst his Neighbours and give him an ill-name, and that's as bad.

The earliest close match to the cited expression in a Hathi Trust search appears in Nathan Bailey, Dictionarium Britannicum: or, A More Compleat Universal Etymological Dictionary (1736):

To him that wills, ways will not be wanting. Lat. Malefacere qui vult, nunquam non causam inveniet. He that is set upon doing another man a mischief, will never want either a pretence or a means to effect it. The Germans say Wer den hund schlagen will finder bald einem knüppel. (He that hath a Mind to strike a Dog, will soon find a Stick.) Or, Give a Dog a bad Name and knock his Brains out; for if you don't another will.That is, easily entangled. Qui veut battre son Chien, trouve assez des Bâtons. Fr.

The tropes of dog hanging and dog slandering also appear together in an "old Land Story" related in William Varon, A Just Defence of the Royal Martyr, K. Charles I, from the Many False and Malicious Aspersions in Ludlow's Memoirs and Some Other Virulent Libels of That Kind (1699):

In the mean while commend me to any Man, or Body of Men, who can have the Confidence to declaim against Arbitrary Power, and yet proceed upon common Fame (which was ever thought hard, and therefore discontinued both in Civil and Canon Law, where for some time it took place) especially in an Age, where Calumny and Slander were so scandalously rise, as no honest Man could escape the devouring Words of their false Tongues. Methinks such a Proceedure as this, has some Affinity with that old Land Story of the Cook serving his Dog, who said he would not hang him, only give him an ill Name; and thereupon threw him into the Street, and cry'd, a Mad Dog; which made all the Rabble of two, as well as four legg'd Curs, fall upon, and worry him to Death.

There are certainly few worse names to give a dog than "mad dog."

Other variants on this expression include (from 1857) "give a dog a bad name, and he will become a bad dog"; (from 1860) "give a dog a bad name, and he is a gone dog"; (from 1865) "give a dog a bad name, and he will probably answer to it on all future occasion"; (from 1869) "give a dog a bad name, and he'll bite you"; (from 1884) "give a dog a bad name, and he will deserve it"; and (from 1902) "give a dog a bad name and he will starve." Curiously, Google Books doesn't find any matches for "Give a dog a bad name and hear [or see or watch] him bark."

I don't whether "give a dog a bad name and hang him" is a direct ancestor of "give a dog a bad name and hear [or see] him bark," but the wording and the sentiment that the two sayings express are close enough to make such a connection quite likely.