To determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for the exponential function [tex]\(f(x) = ab^x\)[/tex] that passes through points [tex]\((0, 7000)\)[/tex] and [tex]\((2, 1120)\)[/tex], we can follow these steps:
1. Point (0, 7000):
When [tex]\(x = 0\)[/tex]:
[tex]\[
f(0) = ab^0 = a \cdot 1 = a
\][/tex]
Since [tex]\(f(0) = 7000\)[/tex], we have:
[tex]\[
a = 7000
\][/tex]
2. Point (2, 1120):
When [tex]\(x = 2\)[/tex]:
[tex]\[
f(2) = ab^2
\][/tex]
Given [tex]\(f(2) = 1120\)[/tex], we substitute [tex]\(a\)[/tex] from the first point:
[tex]\[
1120 = 7000 \cdot b^2
\][/tex]
3. Solve for [tex]\(b^2\)[/tex]:
We need to isolate [tex]\(b^2\)[/tex]. Divide both sides by 7000:
[tex]\[
b^2 = \frac{1120}{7000}
\][/tex]
4. Simplify the fraction:
[tex]\[
b^2 = \frac{112}{700} = \frac{16}{100} = 0.16
\][/tex]
5. Solve for [tex]\(b\)[/tex]:
Take the square root of both sides to find [tex]\(b\)[/tex]:
[tex]\[
b = \sqrt{0.16} = 0.4
\][/tex]
Putting all this together, the values are:
[tex]\[
\begin{array}{l}
a = 7000 \\
b = 0.4
\end{array}
\][/tex]
