To find the exponential model of best fit for the given points (-3, 5), (1, 12), (5, 72), and (7, 137), we start with the general form of an exponential function:
\[ y = ab^x \]
where \( a \) is the initial value (the value of \( y \) when \( x = 0 \)), and \( b \) is the growth factor.
Step 1: Setting Up the Equations
We will use two of the points to set up equations in terms of \( a \) and \( b \). Let's select the points (-3, 5) and (5, 72):
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For the point (-3, 5): \[ 5 = ab^{-3} \] This can be rewritten as: \[ ab^{-3} = 5 \tag{1} \]
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For the point (5, 72): \[ 72 = ab^{5} \] This can be rewritten as: \[ ab^{5} = 72 \tag{2} \]
Step 2: Dividing Equations
Now, divide equation (2) by equation (1) to eliminate \( a \):
\[ \frac{ab^{5}}{ab^{-3}} = \frac{72}{5} \]
This simplifies to:
\[ b^{8} = \frac{72}{5} \]
Step 3: Solving for \( b \)
Calculating the right side:
\[ \frac{72}{5} = 14.4 \]
Now taking the 8th root: \[ b = (14.4)^{1/8} \]
Calculating \( b \):
\[ b \approx 1.402 \]
Step 4: Finding \( a \)
Now substitute \( b \) back into equation (1) to solve for \( a \): \[ 5 = a(1.402)^{-3} \]
Calculating \( (1.402)^{-3} \): \[ (1.402)^{-3} \approx 0.513 \]
Then we can set up the equation: \[ 5 = a \cdot 0.513 \]
Solving for \( a \): \[ a \approx \frac{5}{0.513} \approx 9.75 \]
Step 5: Writing the Exponential Model
Thus the exponential model can be approximated as: \[ y \approx 9.75 \cdot (1.40)^{x} \]
Step 6: Verification
To verify the model, we can plug in other \( x \) values from the data points into the model and see if the corresponding \( y \) values are close to the data points.
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For \( x = 1 \): \[ y \approx 9.75 \cdot (1.40)^{1} \approx 13.65 \quad \text{(close to 12)} \]
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For \( x = 5 \): \[ y \approx 9.75 \cdot (1.40)^{5} \approx 72.89 \quad \text{(close to 72)} \]
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For \( x = 7 \): \[ y \approx 9.75 \cdot (1.40)^{7} \approx 137.70 \quad \text{(close to 137)} \]
Final Model
After verification, the final exponential model of best fit is approximately:
\[ y \approx 9.75(1.40)^{x} \]
All values are rounded to the nearest hundredth as required.