To determine which equation represents an exponential function that passes through the point [tex]\((2, 36)\)[/tex], we need to test each function with the given coordinates [tex]\(x = 2\)[/tex] and [tex]\(y = 36\)[/tex].
### Equation 1: [tex]\( f(x) = 4 \cdot (3)^x \)[/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[ f(2) = 4 \cdot (3)^2 \][/tex]
[tex]\[ f(2) = 4 \cdot 9 \][/tex]
[tex]\[ f(2) = 36 \][/tex]
Since [tex]\(f(2) = 36\)[/tex], we see that the function [tex]\(f(x) = 4 \cdot (3)^x\)[/tex] passes through the point [tex]\((2, 36)\)[/tex].
### Equation 2: [tex]\( f(x) = 4 \cdot x^3 \)[/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[ f(2) = 4 \cdot (2)^3 \][/tex]
[tex]\[ f(2) = 4 \cdot 8 \][/tex]
[tex]\[ f(2) = 32 \][/tex]
Since [tex]\(f(2) = 32\)[/tex], this function [tex]\( f(x) = 4 \cdot x^3 \)[/tex] does not pass through the point [tex]\((2, 36)\)[/tex].
### Equation 3: [tex]\( f(x) = 6 \cdot (3)^x \)[/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[ f(2) = 6 \cdot (3)^2 \][/tex]
[tex]\[ f(2) = 6 \cdot 9 \][/tex]
[tex]\[ f(2) = 54 \][/tex]
Since [tex]\(f(2) = 54\)[/tex], this function [tex]\( f(x) = 6 \cdot (3)^x \)[/tex] does not pass through the point [tex]\((2, 36)\)[/tex].
### Equation 4: [tex]\( f(x) = 6 \cdot x^3 \)[/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[ f(2) = 6 \cdot (2)^3 \][/tex]
[tex]\[ f(2) = 6 \cdot 8 \][/tex]
[tex]\[ f(2) = 48 \][/tex]
Since [tex]\(f(2) = 48\)[/tex], this function [tex]\( f(x) = 6 \cdot x^3 \)[/tex] does not pass through the point [tex]\((2, 36)\)[/tex].
Based on our calculations, the correct equation that represents an exponential function passing through the point [tex]\((2, 36)\)[/tex] is:
[tex]\[ \boxed{f(x) = 4 \cdot (3)^x} \][/tex]
