To determine which points lie on the graph of the function [tex]\( y = \left( \frac{1}{2} \right)^x \)[/tex], let’s evaluate the function for the given [tex]\( x \)[/tex] values and compare the resulting [tex]\( y \)[/tex] values with the ones provided in the points.
1. Point (2, [tex]\(\frac{1}{4}\)[/tex]):
- Substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ y = \left( \frac{1}{2} \right)^2 \][/tex]
- Simplify:
[tex]\[ y = \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) = \frac{1}{4} \][/tex]
- The calculated point is [tex]\((2, \frac{1}{4})\)[/tex], which matches the given point exactly.
2. Point (0, [tex]\(\frac{1}{2}\)[/tex]):
- Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ y = \left( \frac{1}{2} \right)^0 \][/tex]
- Simplify (any number to the power of 0 is 1):
[tex]\[ y = 1 \][/tex]
- The calculated point is [tex]\((0, 1)\)[/tex], which does not match the given point [tex]\((0, \frac{1}{2})\)[/tex].
3. Point (2, 1):
- Substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ y = \left( \frac{1}{2} \right)^2 \][/tex]
- Simplify:
[tex]\[ y = \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) = \frac{1}{4} \][/tex]
- The calculated point is [tex]\((2, \frac{1}{4})\)[/tex], which does not match the given point [tex]\((2, 1)\)[/tex].
4. Point (0, 0):
- Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ y = \left( \frac{1}{2} \right)^0 \][/tex]
- Simplify:
[tex]\[ y = 1 \][/tex]
- The calculated point is [tex]\((0, 1)\)[/tex], which does not match the given point [tex]\((0, 0)\)[/tex].
### Conclusion
The only point which lies on the graph of [tex]\( y = \left( \frac{1}{2} \right)^x \)[/tex] is [tex]\( (2, \frac{1}{4}) \)[/tex].
