What did Jesus mean by his apparent racial taunt to the Canaanite woman in Matt 15 when he inferred that they were "dogs"? In Christs encounter with A woman from Canaan in Matthew 15, why does He infer that the Canaanites are "dogs" and why does the woman apparently accept this racial slur?
There are actually two different Greek words for the English term “dog” that was used in the New Testament. One word is “κύων” (“kýōn”). The actual definition of this word is: “a man of impure mind, an impudent man.” This word would be the word a Jew would use if they were really calling someone else (Jew or Gentile) an “unclean” or unspiritual animal (Matthew 7:6; Luke 16:21; Philippians 3:2). This word was not the word used by Jesus in Matthew 15:26, however.

The word Jesus used was “κυνάριον” (“kynárion”), which simply means “a little dog.” If Jesus truly wanted to insult the Canaanite woman (or Gentiles in general), He would have used the word “κύων,” not “κυνάριον.” 

Also, let’s use a little bit of logic here: if the woman had taken what Jesus said as an insult, her faith would have been damaged and her daughter would not have been healed. She clearly understood Jesus’s reference as nothing more than a harmless metaphor. It’s the same thing as when people are commonly described as “sheep” in the Bible. 

Ultimately, Jesus frequently tested people to prove their intentions, often through response questions or challenges (see John 4:16–18; and 4:50–53). His response to the Canaanite woman is similar. In testing her, Jesus declined her request and explained that she had no legitimate expectation of His help. The woman, however, lived out the principle Jesus Himself taught in the parable of the persistent widow (Luke 18:1–8). Her response proved that she understood fully what Jesus was saying, yet had enough conviction to ask anyway (Matthew 15:27). Jesus acknowledged her faith—calling it “great”—and granted her request (Matthew 15:28). In the end, Jesus happily granted the woman’s request due to her faith.

So, according to both the context and language involved, Jesus wasn’t referring to the Canaanite woman as a “dog,” either directly or indirectly. He wasn’t using an epithet or racial slur but making a point about the priorities He’d been given by God. He was also testing the faith of the woman and teaching an important lesson to His disciples.