To determine the value of [tex]\( y \)[/tex] when the point [tex]\((-3, y)\)[/tex] lies on the graph of the function [tex]\( y = \left( \frac{1}{4} \right)^x \)[/tex], we need to substitute [tex]\( x = -3 \)[/tex] into the given function and solve for [tex]\( y \)[/tex].
Given the function:
[tex]\[ y = \left( \frac{1}{4} \right)^x \][/tex]
We substitute [tex]\( x = -3 \)[/tex]:
[tex]\[ y = \left( \frac{1}{4} \right)^{-3} \][/tex]
Recall that raising a fraction to a negative exponent involves taking the reciprocal of the fraction and then raising it to the positive of that exponent:
[tex]\[ \left( \frac{1}{4} \right)^{-3} = \left( 4 \right)^3 \][/tex]
Next, we calculate [tex]\( 4^3 \)[/tex]:
[tex]\[ 4^3 = 4 \times 4 \times 4 = 64 \][/tex]
Therefore, when [tex]\( x = -3 \)[/tex]:
[tex]\[ y = 64 \][/tex]
So, the point [tex]\( (-3, y) \)[/tex] on the graph results in:
[tex]\[ y = 64 \][/tex]
Thus, the value of [tex]\( y \)[/tex] is:
[tex]\[ \boxed{64} \][/tex]
