Question

Find the exponential model of best fit for the points (-3,5), (1,12), (5,72), (7,137). Explain how you got your answer. Round values to the nearest hundredth.
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Answer
y \approx 13.72(1.4)^{x}


Explanation
The general form of an exponential equation is y = ab^{x}


Use points (-3, 5)
and (5, 72)
to create equations:

5 = ab^{-3}

72 = ab^{5}

Divide the second equation by the first:
\frac{72}{5} = b^{8}



Solve for b
:
b \approx 1.4

Substitute b
back into the first equation to find a
:
5 = a(1.4)^{-3}


Solve for a
:
a \approx 13.72

The exponential model of best fit is y \approx 13.72(1.4)^{x}

Answer 1

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):

1. For the point (-3, 5):
\[
5 = ab^{-3}
\]
This can be rewritten as:
\[
ab^{-3} = 5 \tag{1}
\]

2. 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.

1. For \( x = 1 \):
\[
y \approx 9.75 \cdot (1.40)^{1} \approx 13.65 \quad \text{(close to 12)}
\]

2. For \( x = 5 \):
\[
y \approx 9.75 \cdot (1.40)^{5} \approx 72.89 \quad \text{(close to 72)}
\]

3. 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.