While the term “dog” was a slang term for Gentiles, that’s not necessarily the manner in which Jesus is using it here. The word used here is not simply “dogs” but more precisely means “small dogs” as in a child’s pet.
Jesus and his disciples were constantly met by large crowds and the accompanying exhausting work of ministering to them. Jesus’ ministry was to first and foremost to the people of Israel (Matthew 10:5-6), ministry to others would come after Pentecost. The disciples had faced emergency after emergency and the only reason they were in that region was because they had to flee the crowds in Jewish towns. To go to this woman’s home and heal her daughter would mean others would hear about it and Jesus and the disciples would have to spend days ministering to non-Jewish people of the region. This was not to be the focus of Jesus’ earthly ministry and would have further exhausted Jesus’ disciples.
The woman seems to clearly understand that Jesus is making a point about what are the priorities of his ministry. That is why she responds that even dogs, who are second priority when it comes to getting fed, can still eat something while the main priority of feeding the family is being met.
She would appear to be making a similar request of faith as the Roman centurion. Jesus does not praise her boldness but rather her faith. She perfectly understood Jesus’ analogy and she requested not that Jesus stay and minister (taking time away from his first priority) but rather asked that Jesus heal her daughter with a passing word (the same as passing crumbs falling from a table).
