Sure, let's find the exponential function that passes through the points (5, 5) and (8, 135).
An exponential function can be written in the form:
[tex]\[ y = a \cdot b^x \][/tex]
To find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we can use the given points (5, 5) and (8, 135).
First, rewrite the function for the two points:
1. For (5, 5):
[tex]\[ 5 = a \cdot b^5 \][/tex]
2. For (8, 135):
[tex]\[ 135 = a \cdot b^8 \][/tex]
Now, we have two equations:
[tex]\[
\begin{cases}
5 = a \cdot b^5 \\
135 = a \cdot b^8
\end{cases}
\][/tex]
To solve for [tex]\(a\)[/tex] and [tex]\(b\)[/tex], divide the second equation by the first to eliminate [tex]\(a\)[/tex]:
[tex]\[
\frac{135}{5} = \frac{a \cdot b^8}{a \cdot b^5}
\][/tex]
Simplify the right-hand side:
[tex]\[
27 = b^{8-5} \\
27 = b^3 \\
\][/tex]
Now, solve for [tex]\(b\)[/tex]:
[tex]\[
b = \sqrt[3]{27} \\
b = 3
\][/tex]
With [tex]\(b\)[/tex] known, substitute [tex]\(b\)[/tex] back into one of the original equations to solve for [tex]\(a\)[/tex]:
Using [tex]\(5 = a \cdot 3^5\)[/tex]:
[tex]\[
5 = a \cdot 243 \\
a = \frac{5}{243} \\
a \approx 0.0205761
\][/tex]
So, the values are:
[tex]\[
a \approx 0.0205761 \\
b = 3
\][/tex]
Thus, the exponential function that passes through the points (5, 5) and (8, 135) is:
[tex]\[
y = 0.0205761 \cdot 3^x
\][/tex]
