Let's complete the table of values step-by-step.
Given:
[tex]\[
\begin{array}{|c|c|}
\hline
x & 4^{-x} \\
\hline
-1 & 4 \\
\hline
0 & a \\
\hline
2 & b \\
\hline
4 & c \\
\hline
\end{array}
\][/tex]
We need to find the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
For [tex]\( x = 0 \)[/tex]:
[tex]\[ 4^{-x} = 4^0 \][/tex]
Anything raised to the power of 0 is 1, so:
[tex]\[ a = 4^0 = 1 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ 4^{-x} = 4^{-2} \][/tex]
[tex]\[ b = 4^{-2} = \frac{1}{4^2} = \frac{1}{16} = 0.0625 \][/tex]
For [tex]\( x = 4 \)[/tex]:
[tex]\[ 4^{-x} = 4^{-4} \][/tex]
[tex]\[ c = 4^{-4} = \frac{1}{4^4} = \frac{1}{256} = 0.00390625 \][/tex]
So the complete table is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & 4^{-x} \\
\hline
-1 & 4 \\
\hline
0 & 1 \\
\hline
2 & 0.0625 \\
\hline
4 & 0.00390625 \\
\hline
\end{array}
\][/tex]
Therefore:
[tex]\[ a = 1, \quad b = 0.0625, \quad c = 0.00390625 \][/tex]
