Answer :
To find the exponential function that fits these data points, we will use the general form of an exponential function:
[tex]\[ y = a \cdot b^x \][/tex]
where \( a \) and \( b \) are constants we need to determine.
### Step 1: Determine \( a \)
Given the ordered pair \((0, 4)\):
When \( x = 0 \):
[tex]\[ y = a \cdot b^0 \][/tex]
[tex]\[ y = a \cdot 1 \][/tex]
Thus,
[tex]\[ a = 4 \][/tex]
### Step 2: Determine \( b \)
Next, we use another point to find \( b \). Given the ordered pair \((1, 5)\):
When \( x = 1 \),
[tex]\[ y = a \cdot b^1 \][/tex]
Substitute \( a = 4 \) and \( y = 5 \):
[tex]\[ 5 = 4 \cdot b \][/tex]
Solve for \( b \):
[tex]\[ b = \frac{5}{4} = 1.25 \][/tex]
### Step 3: Verify the Exponential Function
We need to verify the values of \( a \) and \( b \) using the other data points to ensure our exponential function correctly models the data.
Using the function \( y = 4 \cdot 1.25^x \):
- For \( x = 2 \):
[tex]\[ y = 4 \cdot 1.25^2 = 4 \cdot 1.5625 = 6.25 \][/tex]
- For \( x = 3 \):
[tex]\[ y = 4 \cdot 1.25^3 = 4 \cdot 1.953125 = 7.8125 \][/tex]
These calculated values match the given \( y \)-values in the table, confirming our function.
### Conclusion
The exponential function that fits the given data points is:
[tex]\[ y = 4 \cdot 1.25^x \][/tex]
This function is verified by the given points in the table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 4 \\ \hline 1 & 5 \\ \hline 2 & 6.25 \\ \hline 3 & 7.8125 \\ \hline \end{array} \][/tex]
[tex]\[ y = a \cdot b^x \][/tex]
where \( a \) and \( b \) are constants we need to determine.
### Step 1: Determine \( a \)
Given the ordered pair \((0, 4)\):
When \( x = 0 \):
[tex]\[ y = a \cdot b^0 \][/tex]
[tex]\[ y = a \cdot 1 \][/tex]
Thus,
[tex]\[ a = 4 \][/tex]
### Step 2: Determine \( b \)
Next, we use another point to find \( b \). Given the ordered pair \((1, 5)\):
When \( x = 1 \),
[tex]\[ y = a \cdot b^1 \][/tex]
Substitute \( a = 4 \) and \( y = 5 \):
[tex]\[ 5 = 4 \cdot b \][/tex]
Solve for \( b \):
[tex]\[ b = \frac{5}{4} = 1.25 \][/tex]
### Step 3: Verify the Exponential Function
We need to verify the values of \( a \) and \( b \) using the other data points to ensure our exponential function correctly models the data.
Using the function \( y = 4 \cdot 1.25^x \):
- For \( x = 2 \):
[tex]\[ y = 4 \cdot 1.25^2 = 4 \cdot 1.5625 = 6.25 \][/tex]
- For \( x = 3 \):
[tex]\[ y = 4 \cdot 1.25^3 = 4 \cdot 1.953125 = 7.8125 \][/tex]
These calculated values match the given \( y \)-values in the table, confirming our function.
### Conclusion
The exponential function that fits the given data points is:
[tex]\[ y = 4 \cdot 1.25^x \][/tex]
This function is verified by the given points in the table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 4 \\ \hline 1 & 5 \\ \hline 2 & 6.25 \\ \hline 3 & 7.8125 \\ \hline \end{array} \][/tex]
