Scriptures where she can be found: Matthew 15:21-28, Mark 7:24-30
Setting: Tyre & Sidon
This woman was only one of two people in the New Testament that Jesus told they had great faith. Both of those people were Gentiles. They had no knowledge of the Law, just Jesus.
This account is recorded in both Matthew and Mark's Gospel. They both give some different details. I want to focus on what Matthew had to say (but I will mention Mark too.)
Jesus had been in Galilee, but then He left and went to the region of Tyre and Sidon. If you look at the map, it looks like it is kind of out of the way. But Jesus does nothing without purpose, so we know He was traveling that way for a reason.
Mark's account says, "He entered a house and did not want anyone to know it; yet he could not keep his presence secret. In fact, as soon as she heard about him, a woman whose daughter was possessed by an impure spirit came and fell at His feet."-Mark 7:24-25 (NIV)
We see from the beginning that she is very zealous. She came and sought Jesus out.
Matthew's account says she cries out to Him. That is an act of faith. But what is an even bigger statement of faith, is what she says to Jesus.
The first time she addresses Him in Matthew's account, she calls Him "Lord" and "Son of David." She was acknowledging her faith in Him by calling Him Lord, and the fact that He was Messiah, by calling Him "Son of David."
She continues to call Him Lord all throughout this passage of Scripture expressing her faith in Him.
The finale comes to this. Jesus is still engaging her. Nothing He does is without purpose. These discussions He has with people are to bring them to a place of receiving what He has for them.
And then He calls her a dog.
She is so smart, and in faith. Jesus tells her that it is not right that the children's (Israel's) bread to throw it to the dogs (outsiders, Canaanites.) What does she say to Him?
She tells Him that she still knows what He has is FOR HER. She says,
Yes, Lord, yet even the little dogs eat the crumbs which fall from their MASTER'S table.
She ends their dialogue by calling Jesus Master! She recognizes His divinity and authority and expresses her faith in Him to free her daughter.
And He commends her and tells her "Great is your faith!"
And then her daughter is freed. Jesus didn't touch the girl, or even see her. Mark's account tells us when the woman returns home her child was lying in her bed, and the demon was gone.
There is no space and time with faith. Her desire was immediately delivered because of her faith. But not because of her faith in just anything. Because of her faith in JESUS.