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Question 12 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.

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1 answer

Let's address the input-output table and questions step by step.

Input-Output Table:

| x | y | |-----|-------| | -5 | 1600 | | -2 | 200 | | 1 | 25 | | 4 | 3.125 |

A. Exponential Equation

To find the exponential equation that represents the input-output pairs, we will assume that the relationship can be modeled by the equation:

\[ y = ab^x \]

To determine \( a \) and \( b \), we can use any two pairs from the table.

Using the first pair \((-5, 1600)\):

  1. \( 1600 = ab^{-5} \) ---> (1)

Using another pair, say \((-2, 200)\):

  1. \( 200 = ab^{-2} \) ---> (2)

Now you can solve equations (1) and (2) simultaneously to isolate \( a \) and \( b \).

From equation (2): \[ a = \frac{200}{b^{-2}} = 200b^2 \]

Substituting \( a \) in equation (1): \[ 1600 = (200b^2)b^{-5} \] \[ 1600 = 200b^{-3} \] \[ b^{-3} = \frac{1600}{200} = 8 \] \[ b = \frac{1}{2} \]

Now substituting \( b \) back into \( a \): \[ a = 200 \left(\frac{1}{2}\right)^2 = 200 \times \frac{1}{4} = 50 \]

Thus, the exponential equation is:

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

B. Real-World Situation

A potential real-world scenario could involve the decay of a certain type of bacteria in a controlled environment, where \( x \) represents the time in hours and \( y \) is the quantity of bacteria remaining.

C. Explanation of Variables

  • x: Time in hours (the independent variable)
  • y: Quantity of bacteria remaining (the dependent variable)
  • a: Initial quantity of bacteria (at time 0); here, \( a = 50 \).
  • b: The decay factor, which indicates how the quantity of bacteria halves every unit of time (here, \( b = \frac{1}{2} \), meaning the bacteria population halves every hour).

D. Graph for the Table

The graph should plot the points from the table. The curve should start high and descend sharply due to the exponential decay.

To sketch the graph of the exponential equation \( y = 50 \left( \frac{1}{2} \right)^x \):

  1. Create axes where the x-axis is time (in hours) and the y-axis is the quantity of bacteria.
  2. Plot the points corresponding to the x and y from the input-output table.

Assuming you have graphing software or graph paper, your graph will show an exponential decay curve, decreasing as \( x \) increases.

E. Analysis Based on the Exponential Equation and Graph

  • Change in output value per unit change in input value: In this decay scenario, with each increase of 1 in \( x \) (each hour), the output value is reduced to half, indicating a consistent decay rate.
  • What is the output value when the input value is 0:
    When \( x = 0 \), \[ y = 50 \left( \frac{1}{2} \right)^0 = 50 \]
  • What is the output value when the input value is 3:
    When \( x = 3 \), \[ y = 50 \left( \frac{1}{2} \right)^3 = 50 \times \frac{1}{8} = 6.25 \]
  • Prediction based on table and graph: Based on the pattern observed in the table and the graph, if the trend continues, it can be predicted that after 5 hours, the quantity of bacteria will be 50 * (1/2)^5 = ${50}/{32} \approx 1.5625$.

These predictions indicate a rapidly decreasing population which could reflect behavior observed with certain bacterial populations under specific conditions.

Let me know if you need any more help!