To determine which points lie on the graph of the function [tex]\( k(x) = \left( \frac{1}{4} \right)^x \)[/tex], let's evaluate [tex]\( k(x) \)[/tex] at the given [tex]\( x \)[/tex]-values and compare with the corresponding [tex]\( y \)[/tex]-values in each point:
1. Point [tex]\(\left( 3, \frac{1}{64} \right)\)[/tex]:
- Evaluate [tex]\( k(3) \)[/tex]:
[tex]\[
k(3) = \left( \frac{1}{4} \right)^3 = \frac{1}{4^3} = \frac{1}{64}
\][/tex]
- The [tex]\( y \)[/tex]-value we get is [tex]\(\frac{1}{64}\)[/tex]. Thus, [tex]\(\left( 3, \frac{1}{64} \right)\)[/tex] is on the graph.
2. Point [tex]\((-1, 4)\)[/tex]:
- Evaluate [tex]\( k(-1) \)[/tex]:
[tex]\[
k(-1) = \left( \frac{1}{4} \right)^{-1} = \left( \frac{4}{1} \right) = 4
\][/tex]
- The [tex]\( y \)[/tex]-value we get is [tex]\(4\)[/tex]. Thus, [tex]\((-1, 4)\)[/tex] is on the graph.
3. Point [tex]\((1, 4)\)[/tex]:
- Evaluate [tex]\( k(1) \)[/tex]:
[tex]\[
k(1) = \left( \frac{1}{4} \right)^1 = \frac{1}{4}
\][/tex]
- The [tex]\( y \)[/tex]-value we get is [tex]\(\frac{1}{4}\)[/tex], which is not equal to [tex]\(4\)[/tex]. Thus, [tex]\((1, 4)\)[/tex] is not on the graph.
4. Point [tex]\(\left( 0, \frac{1}{4} \right)\)[/tex]:
- Evaluate [tex]\( k(0) \)[/tex]:
[tex]\[
k(0) = \left( \frac{1}{4} \right)^0 = 1
\][/tex]
- The [tex]\( y \)[/tex]-value we get is [tex]\(1\)[/tex], which is not equal to [tex]\(\frac{1}{4}\)[/tex]. Thus, [tex]\(\left( 0, \frac{1}{4} \right)\)[/tex] is not on the graph.
Therefore, the points that are on the graph of [tex]\( k(x) = \left( \frac{1}{4} \right)^x \)[/tex] are:
[tex]\[
\left( 3, \frac{1}{64} \right) \quad \text{and} \quad (-1, 4)
\][/tex]
