Gentiles were despised by the Jews like a dog. These days a dog is known as a good friend to lonely Americans. But in those times, a dog was a symbol of shame and disgrace because he did not wear any clothes. Simply speaking, Jesus was calling her a dog. As we know well, women are very emotional due to their sensitivity.
Most women like to receive admiration, even if it is nothing but lip service.
Women never expect any insult from men. If a woman is insulted, the one who insulted her will never get respect or love from her again. This woman was also a human being and a sensitive woman. She could have gotten very upset when Jesus called her a dog. But, because of her daughter, her heart was broken.