Step 1 :To ensure the function is continuous, the end of one piece must meet the start of the next piece. We can find the values of A, B, C, and D by setting the end of one piece equal to the start of the next piece and solving for the unknown.
Step 2 :For the first piece, the function is 3 when \(x \leq -2\). The next piece starts at \(x = -2\), so we set 3 equal to \(2(-2) + A\) and solve for A: \[3 = -4 + A\] \[A = 3 + 4\] \[\boxed{A = 7}\]
Step 3 :For the second piece, the function is \(2x + A\) when \(-2 < x \leq -1\). The next piece starts at \(x = -1\), so we set \(2(-1) + A\) equal to \(B(-1) + 3\) and solve for B: \[-2 + 7 = -B + 3\] \[\boxed{B = 2}\]
Step 4 :For the third piece, the function is \(Bx + 3\) when \(-1 < x \leq 0\). The next piece starts at \(x = 0\), so we set \(B(0) + 3\) equal to \(6(0) + C\) and solve for C: \[3 = C\] \[\boxed{C = 3}\]
Step 5 :For the fourth piece, the function is \(6x + C\) when \(0 < x \leq 1\). The next piece starts at \(x = 1\), so we set \(6(1) + C\) equal to \(D(1) + 13\) and solve for D: \[6 + 3 = D + 13\] \[\boxed{D = -4}\]
Step 6 :So, the parameters A, B, C, and D are 7, 2, 3, and -4, respectively.