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dang. i got a fifteen-question portfolio for math ive been working on all morning!
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its got it to where each question, has, like, five parts! AAAAGH!!
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Question 9 Use the input-output table to answer the questions. 𝑥 𝑦 −1 5 2 320 5 20,480 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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GPT-4o mini
That sounds like quite a task! How can I help you with your math portfolio? Do you have specific questions or topics you're working on?
Answered by
GPT-4o mini
That does sound overwhelming! Breaking it down into smaller, manageable parts can really help. If you’d like, you can share a few of the questions or parts that you're finding challenging, and I can help you work through them. What do you think?
Answered by
GPT-4o mini
Let’s break down Question 9 step by step.
### A. Write the exponential equation that represents the input-output pairs in the table.
An exponential equation is typically in the form of \( y = ab^x \), where:
- \( a \) is the initial value (output when \( x = 0 \)),
- \( b \) is the growth factor,
- \( x \) is the input.
From the input-output pairs given:
- \( (-1, 5) \)
- \( (2, 320) \)
- \( (5, 20480) \)
We can find \( b \) by using the pairs.
To find \( b \):
1. Using \( (2, 320) \) and \( (5, 20480) \):
\[
\text{From } (2, 320): y = ab^2 \implies 320 = ab^2
\]
\[
\text{From } (5, 20480): y = ab^5 \implies 20480 = ab^5
\]
Dividing the two equations:
\[
\frac{20480}{320} = \frac{ab^5}{ab^2} \implies 64 = b^3 \implies b = 4
\]
Now, substitute \( b \) back into one of the equations to find \( a \):
\[
320 = ab^2 \implies 320 = a(4^2) \implies 320 = 16a \implies a = 20
\]
So, the equation is:
**\( y = 20 \cdot 4^x \)**
### B. Create a real-world situation that could be modeled by the input-output pairs in the table.
**Situation:** A bacterial culture grows exponentially over time. The amount of bacteria present after \( x \) hours is represented by the input-output pairs. At hour -1 (before the culture started), there were 5 bacteria. After 2 hours, there are 320 bacteria, and after 5 hours, there are 20480 bacteria.
### C. Explain what \( x, y, a, \) and \( b \) represent in your real-world situation.
- \( x \): The number of hours elapsed since the start of the observation.
- \( y \): The number of bacteria present at the end of the \( x \) hours.
- \( a = 20 \): The expected initial number of bacteria at time \( x = 0 \) (in this model, it's extrapolated from the data).
- \( b = 4 \): The growth factor, which indicates that the bacteria population quadruples every hour.
### D. Create a graph for the table.
To graph the input-output pairs, plot the points from the table:
- Plot the points: (-1, 5), (2, 320), (5, 20480).
You will have exponential growth, so you would see points rising steeply. You'd want to create a smooth curve that represents the growth showing an increasing trend.
### E. Use the exponential equation and graph to answer the following:
1. **What is the change in the output value per unit change in the input value?**
- The change is not constant; it's exponential. Each hour, the output (bacteria count) becomes four times the previous count.
2. **What is the output value when the input value is 0?**
\[
y = 20 \cdot 4^0 = 20 \cdot 1 = 20
\]
3. **What is the output value when the input value is 3?**
\[
y = 20 \cdot 4^3 = 20 \cdot 64 = 1280
\]
4. **Use the table and graph to predict about the situation you created in part B.**
- Based on the trend in bacterial growth, we can predict that the bacteria will continue to increase exponentially, leading to a very large population if conditions remain favorable. For instance, after 6 hours, we might expect there to be approximately \( 20 \cdot 4^6 = 20 \cdot 4096 = 81920 \) bacteria.
Feel free to modify the real-world scenario or the variables if it doesn't fit your assignment! Let me know if you need further assistance with any part.
### A. Write the exponential equation that represents the input-output pairs in the table.
An exponential equation is typically in the form of \( y = ab^x \), where:
- \( a \) is the initial value (output when \( x = 0 \)),
- \( b \) is the growth factor,
- \( x \) is the input.
From the input-output pairs given:
- \( (-1, 5) \)
- \( (2, 320) \)
- \( (5, 20480) \)
We can find \( b \) by using the pairs.
To find \( b \):
1. Using \( (2, 320) \) and \( (5, 20480) \):
\[
\text{From } (2, 320): y = ab^2 \implies 320 = ab^2
\]
\[
\text{From } (5, 20480): y = ab^5 \implies 20480 = ab^5
\]
Dividing the two equations:
\[
\frac{20480}{320} = \frac{ab^5}{ab^2} \implies 64 = b^3 \implies b = 4
\]
Now, substitute \( b \) back into one of the equations to find \( a \):
\[
320 = ab^2 \implies 320 = a(4^2) \implies 320 = 16a \implies a = 20
\]
So, the equation is:
**\( y = 20 \cdot 4^x \)**
### B. Create a real-world situation that could be modeled by the input-output pairs in the table.
**Situation:** A bacterial culture grows exponentially over time. The amount of bacteria present after \( x \) hours is represented by the input-output pairs. At hour -1 (before the culture started), there were 5 bacteria. After 2 hours, there are 320 bacteria, and after 5 hours, there are 20480 bacteria.
### C. Explain what \( x, y, a, \) and \( b \) represent in your real-world situation.
- \( x \): The number of hours elapsed since the start of the observation.
- \( y \): The number of bacteria present at the end of the \( x \) hours.
- \( a = 20 \): The expected initial number of bacteria at time \( x = 0 \) (in this model, it's extrapolated from the data).
- \( b = 4 \): The growth factor, which indicates that the bacteria population quadruples every hour.
### D. Create a graph for the table.
To graph the input-output pairs, plot the points from the table:
- Plot the points: (-1, 5), (2, 320), (5, 20480).
You will have exponential growth, so you would see points rising steeply. You'd want to create a smooth curve that represents the growth showing an increasing trend.
### E. Use the exponential equation and graph to answer the following:
1. **What is the change in the output value per unit change in the input value?**
- The change is not constant; it's exponential. Each hour, the output (bacteria count) becomes four times the previous count.
2. **What is the output value when the input value is 0?**
\[
y = 20 \cdot 4^0 = 20 \cdot 1 = 20
\]
3. **What is the output value when the input value is 3?**
\[
y = 20 \cdot 4^3 = 20 \cdot 64 = 1280
\]
4. **Use the table and graph to predict about the situation you created in part B.**
- Based on the trend in bacterial growth, we can predict that the bacteria will continue to increase exponentially, leading to a very large population if conditions remain favorable. For instance, after 6 hours, we might expect there to be approximately \( 20 \cdot 4^6 = 20 \cdot 4096 = 81920 \) bacteria.
Feel free to modify the real-world scenario or the variables if it doesn't fit your assignment! Let me know if you need further assistance with any part.