To determine the value of [tex]\( y \)[/tex] for the given point [tex]\((-2, y)\)[/tex] which lies on the graph of [tex]\( y = 4^x \)[/tex], follow these steps:
1. Identify the given information:
- The point in question is [tex]\((-2, y)\)[/tex].
- The equation of the curve is [tex]\( y = 4^x \)[/tex].
2. Substitute the x-coordinate into the equation:
- Since the x-coordinate is [tex]\(-2\)[/tex], substitute [tex]\( x = -2 \)[/tex] into the equation [tex]\( y = 4^x \)[/tex].
[tex]\[
y = 4^{-2}
\][/tex]
3. Simplify the expression:
- Rewrite the negative exponent as a positive one by taking the reciprocal.
[tex]\[
4^{-2} = \frac{1}{4^2}
\][/tex]
4. Calculate the power of 4:
- Compute [tex]\( 4^2 \)[/tex].
[tex]\[
4^2 = 16
\][/tex]
5. Find the reciprocal:
- Take the reciprocal of 16.
[tex]\[
\frac{1}{4^2} = \frac{1}{16}
\][/tex]
6. Conclusion:
- The value of [tex]\( y \)[/tex] when [tex]\( x = -2 \)[/tex] is:
[tex]\[
y = \frac{1}{16}
\][/tex]
Therefore, if [tex]\((-2, y)\)[/tex] lies on the graph of [tex]\( y = 4^x \)[/tex], then [tex]\( y = \frac{1}{16} \)[/tex]. The correct answer is:
[tex]\[
\boxed{\frac{1}{16}}
\][/tex]
