AP Precalculus Practice Test: Unit 2 Question #14 TI-84+ Calculator to Do ExpReg

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### AP Precalculus Practice Test: Unit 2, Question #14
**Topic:** Using the TI-84+ Calculator to Perform Exponential Regression (ExpReg)

This question involves using the TI-84+ calculator to find the exponential regression (ExpReg) for a set of data. Exponential regression is a statistical method used to fit an exponential curve to a set of data points. This method is useful for modeling situations where growth or decay follows an exponential pattern.

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### **Step-by-Step Instructions for Using the TI-84+ Calculator to Perform Exponential Regression (ExpReg):**

#### 1. **Enter Data into Lists:**
- Turn on the calculator and press the **`STAT`** button.
- Select **1: Edit** and press **`ENTER`** to enter the data editor.
- Input the \(x\)-values into **List 1** (L1) and the corresponding \(y\)-values into **List 2** (L2).
- For example, if you have the following data points:
\[
(1, 2), (2, 4), (3, 8), (4, 16)
\]
- Enter the \(x\)-values \(1, 2, 3, 4\) into **L1**, and the \(y\)-values \(2, 4, 8, 16\) into **L2**.

#### 2. **Access the Exponential Regression Function:**
- Press the **`STAT`** button again to open the statistics menu.
- Use the arrow keys to scroll right to **CALC** (calculation options) and then select **ExpReg** (Option 0). This is the exponential regression function.
- The ExpReg command is used to fit an exponential model of the form:
\[
y = ab^x
\]
where \(a\) is the initial value (when \(x = 0\)) and \(b\) is the base that describes the growth or decay rate.

#### 3. **Perform the Regression:**
- After selecting **ExpReg**, the calculator will display:
\[
\text{ExpReg} \quad Y1 = a \cdot b^X
\]
- It should automatically use **L1** for the \(x\)-values and **L2** for the \(y\)-values. If it doesn't, you can manually input **L1** and **L2** by typing them after the command.
- Press **`ENTER`** to calculate the regression. The calculator will display the values of \(a\) and \(b\).

#### 4. **Interpret the Results:**
- The calculator will output the values for \(a\) and \(b\), which define the exponential function that best fits your data.
- For example, the result might be something like:
\[
y = 1.5 \cdot 2^x
\]
- Here, \(a = 1.5\) and \(b = 2\), so the exponential model for this data is \(y = 1.5 \cdot 2^x\).

#### 5. **Graph the Exponential Model (Optional):**
- To graph the exponential function, press the **`Y=`** button to enter the function editor.
- Input the exponential model (e.g., \(Y1 = 1.5 \cdot 2^X\)) into the equation editor.
- Press **`GRAPH`** to display the graph of the exponential function along with the data points.

#### 6. **Use the Model to Make Predictions:**
- You can use the exponential model \(y = ab^x\) to predict future values or understand the behavior of the system represented by the data.
- To do this, simply substitute a value for \(x\) into the equation and solve for \(y\).

---

### **Example Problem:**

Suppose you are given the following data points, which represent the population of a species over time (in years):

| Year (x) | Population (y) |
|----------|----------------|
| 0 | 100 |
| 1 | 150 |
| 2 | 225 |
| 3 | 337.5 |

You are asked to find the exponential model that best fits this data using the TI-84+ calculator.

---

### **Solution:**

1. **Enter the data:**
- Press **`STAT`**, then **1: Edit**.
- Enter the \(x\)-values \(0, 1, 2, 3\) into **L1**.
- Enter the corresponding \(y\)-values \(100, 150, 225, 337.5\) into **L2**.

2. **Perform Exponential Regression (ExpReg):**
- Press **`STAT`**, then scroll right to **CALC** and select **ExpReg** (Option 0).
- The calculator will automatically use **L1** and **L2**. Press **`ENTER`** to calculate.

3. **Interpret the output:**
- Suppose the calculator outputs:
\[
y = 100 \cdot 1.5^x
\]
- This means the exponential model is \(y = 100 \cdot 1.5^x\), where \(a = 100\) and \(b = 1.5\).

4. **Graph the model (Optional):**
- Press **`Y=`** and enter the equation \(Y1 = 100 \cdot 1.5^X\).
- Press **`GRAPH`** to see the graph of the model.

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Nick Perich
Norristown Area High School
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