To find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for the exponential function [tex]\(h(x) = a \cdot b^x\)[/tex] that fits the given data points, let's break it down step by step using the information provided in the table:
[tex]\[
\begin{array}{cc}
x & h(x) \\
\hline
0 & 10 \\
1 & 4 \\
\end{array}
\][/tex]
1. Determine [tex]\(a\)[/tex]:
When [tex]\(x = 0\)[/tex], we have:
[tex]\[
h(0) = a \cdot b^0
\][/tex]
Since [tex]\(b^0 = 1\)[/tex], this simplifies to:
[tex]\[
h(0) = a \cdot 1 \implies a = h(0) = 10
\][/tex]
2. Determine [tex]\(b\)[/tex]:
When [tex]\(x = 1\)[/tex], we have:
[tex]\[
h(1) = a \cdot b^1
\][/tex]
Substituting the values from the table and [tex]\(a = 10\)[/tex], this becomes:
[tex]\[
4 = 10 \cdot b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = \frac{4}{10} = 0.4
\][/tex]
Therefore, the exponential function can be written as:
[tex]\[
h(x) = 10 \cdot (0.4)^x
\][/tex]
So, the completed equation for [tex]\(h(x)\)[/tex] is:
[tex]\[
h(x) = 10 \cdot (0.4)^x
\][/tex]
