To determine the exponential function from the given table, we need a function of the form [tex]\( y = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
We will use the given pairs [tex]\((x, y)\)[/tex] to find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
Given table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 1.5 \\
\hline
1 & 4.5 \\
\hline
2 & 13.5 \\
\hline
3 & 40.5 \\
\hline
\end{array}
\][/tex]
### Step 1: Determine the value of [tex]\( a \)[/tex]
When [tex]\( x = 0 \)[/tex], the equation becomes:
[tex]\[ y = a \cdot b^0 \][/tex]
Since [tex]\( b^0 = 1 \)[/tex], this simplifies to:
[tex]\[ y = a \][/tex]
From the table, when [tex]\( x = 0 \)[/tex], [tex]\( y = 1.5 \)[/tex]. Therefore,
[tex]\[ a = 1.5 \][/tex]
### Step 2: Determine the value of [tex]\( b \)[/tex]
Next, let's use the second pair [tex]\((1, 4.5)\)[/tex] to find [tex]\( b \)[/tex].
When [tex]\( x = 1 \)[/tex], the function is:
[tex]\[ y = a \cdot b^1 \][/tex]
Substituting the values we know:
[tex]\[ 4.5 = 1.5 \cdot b \][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[ b = \frac{4.5}{1.5} = 3 \][/tex]
### Step 3: Verify the function with other points
Now, let's verify the function [tex]\( y = 1.5 \cdot 3^x \)[/tex] with the remaining points:
- For [tex]\( x = 2 \)[/tex], we get:
[tex]\[ y = 1.5 \cdot 3^2 = 1.5 \cdot 9 = 13.5 \][/tex]
This matches the table value.
- For [tex]\( x = 3 \)[/tex], we get:
[tex]\[ y = 1.5 \cdot 3^3 = 1.5 \cdot 27 = 40.5 \][/tex]
This also matches the table value.
Therefore, the function that fits the given data is:
[tex]\[ y = 1.5 \cdot 3^x \][/tex]
