Question

Use the input-output table to answer the questions.
𝑥 𝑦
−5 1600
−2 200
1 25
4 3.125
A. Write the exponential equation that represents the input-output pairs in the
table.
B. Create a real-world situation that could be modeled by the input-output pairs
in the table.
C. Explain what x, y, a, and b represent in your real-world situations.
D. Create a graph for the table.
E. Use the exponential equation and graph to answer the following:
• What is the change in the output value per unit change in the inputvalue?
• What is the output value when the input value is 0?
• What is the output value when the input value is 3?
• Use the table and graph to make a prediction about the situation you
created in part B.
Make the real world situation about a nail saloon

Answer 1

Let's address each part of your request step-by-step.

### A. Write the exponential equation that represents the input-output pairs in the table.

To derive the exponential equation from the input-output pairs, we first recognize the typical form of an exponential function:

\[ y = ab^x \]

From the table, we can use the pairs to find \( a \) and \( b \). It appears the points suggest a decay or growth pattern.

We can try to take two of the points, such as:

- Point 1: (-5, 1600)
- Point 2: (4, 3.125)

1. **For the point (-5, 1600):**
\[
1600 = ab^{-5} \quad \text{(1)}
\]

2. **For the point (4, 3.125):**
\[
3.125 = ab^{4} \quad \text{(2)}
\]

Now we have two equations. Dividing equation (1) by equation (2):

\[
\frac{1600}{3.125} = \frac{ab^{-5}}{ab^{4}} \implies \frac{1600}{3.125} = b^{-9}
\]

Calculating \( \frac{1600}{3.125} \):

\[
\frac{1600}{3.125} = 512
\]

Thus,

\[
b^{-9} = 512 \implies b = 512^{-1/9}
\]

From the properties of exponents, we find:

\[
b = \frac{1}{2}
\]

Now substitute \( b \) back to find \( a \) using either equation. Let's use equation (1):

\[
1600 = a\left(\frac{1}{2}\right)^{-5} \implies 1600 = a \cdot 32 \implies a = 50
\]

Thus, the exponential equation is:

\[
y = 50\left(\frac{1}{2}\right)^{x}
\]

### B. Create a real-world situation that could be modeled by the input-output pairs in the table.

**Situation:** A nail salon uses a specific type of nail polish that decreases in inventory as the number of customers (denoted by \( x \)) increases. Starting at an inventory of 1600 bottles when no customers have come in (negative input values), the inventory decreases as each customer uses some nail polish for their service.

### C. Explain what x, y, a, and b represent in your real-world situations.

- **\( x \)**: This represents the number of customers who have visited the nail salon. It can be negative, indicating a baseline inventory before customers arrived.
- **\( y \)**: This symbolizes the amount of nail polish (in bottles) remaining in the salon's inventory.
- **\( a \)**: This is the starting amount of nail polish (50 bottles when the inventory is considered in relation to the number of customer visits).
- **\( b \)**: This represents the rate at which the inventory decreases per customer. Since \( b = \frac{1}{2} \), it shows that for every customer, the inventory drops significantly, reflecting a high turnover of nail polish.

### D. Create a graph for the table.

To graph the given data points, evaluate the points calculated from the input-output table. The axes would be labeled as follows:

- **x-axis**: Number of Customers (x)
- **y-axis**: Inventory of Nail Polish Bottles (y)

Here are the points to plot:
- (-5, 1600)
- (-2, 200)
- (1, 25)
- (4, 3.125)

(You would need to graph these points using graph paper or software, showing the exponential decay.)

### E. Use the exponential equation and graph to answer the following:

**What is the change in the output value per unit change in the input value?**
- The output decreases exponentially as the input increases. Thus, there is not a constant change in the output per unit increase in input.

**What is the output value when the input value is 0?**
\[
y = 50\left(\frac{1}{2}\right)^{0} = 50\cdot1 = 50
\]

**What is the output value when the input value is 3?**
\[
y = 50\left(\frac{1}{2}\right)^{3} = 50\cdot\left(\frac{1}{8}\right) = 6.25
\]

**Use the table and graph to make a prediction about the situation you created in part B.**
- The prediction could state that if the nail salon has more than 3 customers, the inventory of the nail polish will be reduced to a critical level. For example, with 4 or more customers, the salon may run very low on stock and could face a shortage of nail polish.

This outlines all aspects as requested, specific to the nail salon scenario!