The unity and the sign of the covenant of grace accross both testaments

Here's a good info-graphic, explaining the covenant of grace. We should read the new covenant in context of the Old Testament. The New Covenant is new, not in making absolute the previous covenants. It is new in bringing to fulfilment the promises made before. The New Covenant is built on the Noahic, Abrahamic, Mosaic, and Davidic covenants with Christ as the foundation of them all. The same principle runs through them all (dotted line): Salvation by grace alone, through faith alone, in Christ alone, sealed with a covenant sign.

If this is true, where does this place the children of believers? Well, God said to Abraham: "I will be a God to you and your descendants" (Gen. 17:7) From creation, family was a covenant unit. Adam and Eve were the first covenant community. And although they sinned, God extended grace through the first sacrifice. He also introduced the covenant of grace with Christ as the substance (Gen. 3:15). All subsequent covenants are built on the covenant of grace with Christ as the substance. All of them from Noah had a sign. With Abraham it took on more form as we know it today. The children of believers were seen as covenant children of God and received the covenant sign (Gen. 17).

We see the same principles continuing in the New Testament (Acts 2:39; 1 Cor. 7:14). The New Covenant is the covenant of grace under a new administration but with the same principles. Of that we see examples of the household baptisms in the New Testament (Acts 16:15; 16:33; 1 Cor. 1:16). Many biblical passages also connect household and children (e.g., Gen. 18:19; 31:41; 36:6; 47:12; Num. 18:11; 1 Chron. 10:6; Matt. 19:29; 1 Tim. 3:12), and several others reference entire households being saved (Luke 19:9; John 4:53; Acts 11:14; 16:31; 18:18).

Adam and Eve were the first covenant community. Household baptisms thus speak of the same principle: "I will be a God to you and your children." God never excluded the children of believers as we move into the New Covenant. If fact, the New Covenant was not a reduction, but an expansion to include even the Gentiles. Even now, females were baptized as they could not receive the Old Testament covenant sign. The main principles we should therefore take note of: God has never changed his mind about the children of believers. They are included in the New Covenant because it is the same covenant of grace under which the Abrahamic covenant functioned.