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