The purpose of this essay is to look into the work and life of Jesus Christ while on earth as portrayed by each of the Gospels. Though each of the gospels presents a full picture of the person of Christ however, there is a distinction between the four. Each of the writers presents Jesus in his own way, in terms of audience, theme, and focus.

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The Gospel of Matthew was  written around 50-60AD in Palestine, before the destruction of Jerusalem in 70AD (Matt 4:5, 27:53). Although the specific date for the book of Mark is uncertain, scholars date it to be written around 49s – 60s AD, and believed to be the first Gospel written. The Gospel of Luke was written between 60 – 65AD while the Gospel of John was dated about 90s.
Theme
Matthew presents  Jesus as Messiah. Mark presents Jesus as Servant of God, Luke presents Jesus as the Divine Saviour, and John presents Jesus as the Son of God.
Audience
Matthew writes his gospel to the Jews. He sought to convince the Jews that Jesus is the promised messiah and king, which was fulfilled through repeated prophecy in the Old Testament. Mark writes to the Roman audience, who were not  religious, neither philosopher like the Greek. Luke writes to a Greek audience, who were very religious, philosophical, and great thinkers. His two literal classical books, book of Acts and Luke were written to Theophilus a high ranking Greek noble man and other Gentile Christians (LK 1:3, Acts 1:1). The gospel of John was written to all Christians.
Background to the Gospel
The book of Matthew was placed first as an introduction to the New Testament because it bridge gap between the Old and New Testament. Written in Greek for Jewish believers but the style of writing was Hebrew or Aramaic due to the way Jesus life and ministry was narrated. Although the author of the gospel was not identified by name in biblical text, but testimony by early  church fathers, Eusebius and Papias shows that Apostle Matthew wrote the book as an eye witness to major events during the ministry of Jesus Christ on earth.
The gospel of Mark was action parked fast moving narration written by John Mark an associate of Peter, companion of Paul and Barnabas.  According to Papias, Mark derived his content from his association with Peter, he was Peter interpreter and wrote as many things he can remembered as Peter relate to him. He tailored his gospel for his Romans listeners who are not religious, who have respect for authority but love action. He portrayed Jesus as a miracle worker; one who could control wind and storm, whom demons trembled at his sight and heal all manner of sicknesses and diseases, but despite this power he humbled himself and gave his life as a ransom, teaching that people who are truly great are servant, that anyone that wants to be great must be a servant first.
The author of the gospel of  Luke was a physician, astute writer, theologian and historian whose book was directed to the Greek believer. The book of Luke is the first of its two books addressed to a man called Theophilus a gentile believer (LK 1:3, Act 1:1). He was a Gentile convert, the only non Jewish author of a bible book. Besides his medical knowledge which he shows frequently when he used medical terms to describe sicknesses for example a man suffering from dropsy (LK 14:2).
The author of the fourth gospel was  John the son of Zebedee one of the original twelve disciples of Jesus, he writes as the disciples whom Jesus loved (Jn. 13: 23-24, 20: 2-9, 21:2-24), hence claims to be an eye witness of Jesus ministry, and recorded much of Jesus ministry in Judea and Jerusalem.
Purpose of the Canonical Gospels
Matthew wrote to convinced Jewish Christians that Jesus roots is from the lineage of Abraham and descendants of David (Matt 1:1-17). Also too assured the Jews that Jesus was the son of God, awaited messiah has foretold by the Old Testament prophet (Ish. 7:14).  Matthew also shows to the Jews that Christ came to establish a spiritual kingdom not a political one, which will be followed by a physical kingdom at the end of the age. The book of Matthew is not complete without his explanation on great commission (Matt 28:19).
Marks writes to persecuted  Romans Christian presenting Jesus as the Son of God (Mk 1:1), a suffering servant of God, also too encouraged Romans believers to suffered faithfully for the gospel as he placed before them the life, suffering, death, and resurrection of Jesus Christ their Lord and saviour.
Luke wrote to the  Greek Christian and the most comprehensive of the synoptic gospel, he provide full details of all that Jesus the human saviour had done, teach until the day he was taken up to heaven (LK 1:3-4). He traced Jesus human genealogy back to Adam (LK 3:23-38), not just to Abraham as did Matthew (Matt 1:1-17). In Luke we see Jesus as divine saviour (Lk. 1:47, 2:11) who provides salvation for all men irrespective of their nationality.
Johns purpose was to reveal the  deity of Jesus Christ that he is indeed the Son of God who in obedience to the father took on the nature of man (Jn. 1:1), and those who believed in him receive eternal life (John 20:31). He wrote to refute false teachings of the  Gnostics who goes about denying the humanity: (the word became flesh, Jn. 1:14) and death of Jesus Christ: (Jesus death and resurrection, Jn. 19, 20:1-28).
Focus of the Gospels
Matthew  focus on Jesus, the King of Israel who fulfilled Old Testament laws and messianic prophecy (Ish. 7:13), from conception (Matt 1:22-23), to his residence in Nazareth (Matt 2:23), his Galilean ministry (Matt 4:14-chs 18), teachings about the end time (eschatology), triumphant entry into Jerusalem. He arranged Jesus teachings about the kingdom into; ethics-sermon on the mount (Matt 5:1-7:29); missionary (Matt 9:35-11:1); parables about the kingdom (Matt 13:1-52); discipleship (Matt 18:1-35) and lastly on Olivet discourse about the end of the age (Matt 23:1-25:46). He portrayed Jesus as the promised King, who came to inaugurate kingdom of heaven which is not political but spiritual.
In a fast  catchy action, Mark focuses primarily on Jesus mighty miracles, his authority over sicknesses and demons as signs that the kingdom of Gods is at hand. He also portrayed Jesus as Gods servant. It was at Caesarea Philippi, Jesus identity was fully disclosed to his disciples as the Christ, and his prediction about his death and resurrection (Mk 8:31) which shows his divine foreknowledge; making references to cost of discipleship (Mk 8:34-38, 9:1-10) and lastly on Passion Week in (Mk 11-16).
Luke is the most comprehensive of the synoptic gospel, focusing on Jesus lineage from pre birth to ascension, his  Galilean ministry in LK 4:14-9:50, final journey to Jerusalem in LK 9:51-19:27 and his last week called passion week in LK 19:28-24:53. The main focus of Luke is on Jesus teachings and parables during his ministry, expressing universal hope that Jesus came for both Jews and gentiles, his emphasis on prayer, on the move and work of the Holy Spirit, starting from the filling of John the Baptist from his mother womb; to the miraculously birth of the son of God (LK 1:35), to Jesus spirit baptism in (LK 4:14); and clothing of disciples with power from on high (LK 24:49).
John focuses much more on the discourses of Christ as Gods incarnate Son. The  book of John have two divisions the first division (Jn. 1-12), he explain with evidences Jesus incarnation with: seven signs (miracles), seven teachings (Jesus used to disclose his identity), and seven “I am” statements (by which Jesus revealed figuratively what he is to human race) with his bodily resurrection as proof of his claim as Christ, the Son of God. The second division John focuses on Jesus relationship with his disciples (Jn. 13-21), the ministry of the Holy Spirit (Jn. 14-16), the last supper (Jn. 13), Jesus final prayer for believers (Jn. 17), finally his death and resurrection (Jn. 20-21).
Relation of the Gospels
The four gospel writers write to people of different background and needs. Each under the influence and the inspiration of the Holy Spirit write about the life and teaching of Jesus Christ according to audiences whom they were writing to. Other teaching was omitted purposely, for example Mark mentions women not divorcing their husbands to his Romans audience (Mk 10; 12), Matthew did not mention this since Jewish law did not permit a woman to divorce her husband. Matthew and Luke  write from Mark while John copied from no one. Altogether Matthew, Mark and Luke record eighty-three events of the same account only twenty four of this are  found in all the four gospels, with forty- one accounts unique to Johns Gospel, fifty-two to Luke, Matthew with twenty —seven unique accounts, while Mark has two. John has no record of Jesus parables. Matthew recorded eighteen parables of Jesus, Luke nineteen while Mark has four.
Conclusion
Each of the Gospels writers presents the same gospel under the leading of the Holy Spirit too meet specific need in the body of Christ. I believe the focus of each of the writer matured the church of Christ over time, keep the church focus in the right perspective.