y = √a x² + 3^(b+x) - c

passes through (-2,8) (1,13) and (3,285)


Sub (-2,8),

8 = √a(-2)² + 3^(b - 2) - c
8 = 4√a + 3^(b-2) - c
8 = 4√a + 1/9 (3^b) - c ①


Sub (1,13),
13 = √a + 3^(b+1) - c
13 = √a + 3(3^b) - c ②


Sub (3,285)
285 = √a(3²) + 3^(b + 3) - c
285 = 9√a + 27(3^b) - c ③


② - ① :
5 = -3√a + 26/9 (3^b) ④




③ - ② :

272 = 8√a + 24(3^b)
Multiply by 3/8,

102 = 3√a + 9(3^b) ⑤


⑤ + ④ :

107 = 107/9 (3^b)
3^b = 9 = 3²

So b = 2


Sub 3^b = 9 into ⑤,

102 = 3√a + 9(9) = 3√a + 81
3√a = 21
√a = 7
a = 7² = 49


Sub √a = 7 and 3^b = 9 into ②,

13 = 7 + 3(9) - c = 34 - c
c = 21




Or, use matrix method.

Rearranging ① , ② and ③ gives :

4√a + 1/9 (3^b) - c = 8
√a + 3(3^b) - c = 13
9√a + 27(3^b) - c = 285

Rewrite into matrix form

[4 1/9 -1] [√a] [8]
[ 1 3 -1] [3^b] = [13]
[ 9 27 -1] [c ] [285]


Either find the inverse matrix manually, then use the property

AX = B
X = A‾¹ B to solve. X is your 3 x 1 matrix, A is your 3 x 3 matrix


Or use GC to solve the matrix under the system of linear equations solver app.
(GC will take √a as x , 3^b as y and c as z)



You will get √a = 7, 3^b = 9, c = 21
so a = 7² = 49, and b = 2 since 3^b = 3²