Model the data using an exponential function *f*(*x*) = *Abx*.

*x*

1

2

*f*(*x*)

−59

−87.025

*f*(*x*) = 

To model the given data using an **exponential function** of the form:  
\[
f(x) = A \cdot b^x
\]
we need to determine the parameters \(A\) and \(b\) based on the two data points:  

- \(f(1) = -59\)  
- \(f(2) = -87.025\)

### Step 1: Write two equations  
Using the general form \(f(x) = A \cdot b^x\) for the given points:  
1. For \(x = 1\):
   \[
   -59 = A \cdot b^1 \quad \Rightarrow \quad -59 = A \cdot b
   \]

2. For \(x = 2\):
   \[
   -87.025 = A \cdot b^2
   \]

### Step 2: Eliminate \(A\)  
From the first equation, solve for \(A\):
\[
A = \frac{-59}{b}
\]

### Step 3: Substitute into the second equation  
Plug this value of \(A\) into the second equation:
\[
-87.025 = \left(\frac{-59}{b}\right) \cdot b^2
\]

Simplify:
\[
-87.025 = -59 \cdot b
\]

Solve for \(b\):
\[
b = \frac{-87.025}{-59} = 1.475
\]

### Step 4: Find \(A\)  
Now that we know \(b = 1.475\), substitute this value back into the first equation to find \(A\):
\[
-59 = A \cdot 1.475 \quad \Rightarrow \quad A = \frac{-59}{1.475} \approx -40
\]

### Step 5: Write the Model  
Now that we have \(A \approx -40\) and \(b \approx 1.475\), the exponential function becomes:
\[
f(x) = -40 \cdot (1.475)^x
\]

---

Do you want further clarification or additional steps?  

Here are five related questions:  
1. How do we validate if this model fits the given data well?  
2. Can exponential functions model negative values reliably?  
3. What happens if we shift the base \(b\)?  
4. How do logarithms help solve for unknowns in exponential models?  
5. How does changing \(A\) affect the graph of the function?  

**Tip:** Exponential models are often sensitive to small changes in data—small errors in parameter estimation can lead to noticeable deviations.

Model the data using an exponential function f(x) = Ab^x.

x
1
2
f(x)
−59
−87.025

Learn how to model data using the exponential function f(x) = A * b^x with a step-by-step solution. This problem involves data points (x = 1, f(x) = -59) and (x = 2, f(x) = -87.025), and includes concepts from algebra and exponential growth. Suitable for Grades 10-12.

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