To answer the questions using the input-output table you provided, we'll first determine the exponential relationship, create a real-world scenario, explain the variables involved, create a graph, and then analyze the function in relation to the questions provided.
A. Exponential Equation
The general form of an exponential function is:
\[ y = a \cdot b^x \]
Using the points given in the input-output table:
- When \( x = -1 \), \( y = 5 \)
- When \( x = 2 \), \( y = 320 \)
- When \( x = 5 \), \( y = 20,480 \)
We can find \( a \) and \( b \) using the points.
Let's start by using the first two points to derive the constants \( a \) and \( b \).
Using \( x = -1 \) and \( y = 5 \): \[ 5 = a \cdot b^{-1} \] \[ 5 = \frac{a}{b} \] \[ a = 5b \] ... (1)
Using \( x = 2 \) and \( y = 320 \): \[ 320 = a \cdot b^{2} \] Substituting equation (1) into this: \[ 320 = (5b) \cdot b^{2} \] \[ 320 = 5b^{3} \] \[ b^{3} = 64 \] \[ b = 4 \]
Now substituting \( b \) back into equation (1): \[ a = 5b = 5 \cdot 4 = 20 \]
Thus, the exponential equation will be: \[ y = 20 \cdot 4^x \]
B. Real-World Situation
Letβs assume that \( x \) represents the number of years since a certain product was introduced, and \( y \) represents the total sales (in units) of that product.
For example, a new tech gadget was launched in year 0, and its sales growth over the years follows the described pattern.
C. Explanation of Variables
- \( x \): Represents the number of years since the product was launched.
- \( y \): Represents the total units sold of the product.
- \( a \): The initial sales of the product when it was first launched (at year 0). In this case, \( a = 20 \) units.
- \( b \): The growth factor of sales per year. In this case, \( b = 4 \), meaning the sales quadruple each year.
D. Graph Creation
To graph the equation \( y = 20 \cdot 4^x \), you will plot the following points based on values in the table:
- For \( x = -1 \), \( y = 5 \)
- For \( x = 2 \), \( y = 320 \)
- For \( x = 5 \), \( y = 20,480 \)
Graph these points and also sketch the general behavior of the exponential function that grows rapidly.
E. Analysis using the Exponential Equation and Graph
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Change in Output per Unit Change in Input:
- The output value increases by a factor of \( b = 4 \) for each unit increase in \( x \).
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Output value when the input value is 0: \[ y = 20 \cdot 4^0 = 20 \cdot 1 = 20 \] The output value at \( x = 0 \) is 20.
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Output value when the input value is 3: \[ y = 20 \cdot 4^3 = 20 \cdot 64 = 1280 \] The output value at \( x = 3 \) is 1280.
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Prediction: Based on the exponential growth pattern, it can be predicted that as years increase, the sales in units will continue to rise substantially. For example, by year 4, sales might reach over 5120 units.
This structured analysis captures the essence of exponential growth in a real-world scenario, facilitating understanding and prediction.
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