Question
The first point of the parabola is at (1, 14), the second point is at (2, 4).
Given the graph of the exponential equation, which option is the correct exponential equation for the graph?
1. y = 7(1/2)^x
2. 49(2/7)^x
3. y = 7(2/7)^x
4. y = 49(7/2)^x
Given the graph of the exponential equation, which option is the correct exponential equation for the graph?
1. y = 7(1/2)^x
2. 49(2/7)^x
3. y = 7(2/7)^x
4. y = 49(7/2)^x
Answers
GPT-4o mini
To determine the correct exponential equation for the graph based on the given points (1, 14) and (2, 4), we can start by evaluating each option to see which one fits both points.
Let’s denote the exponential general form as \( y = ab^x \), where \( a \) is the initial value (when \( x=0 \)) and \( b \) is the base of the exponent.
### Evaluation of the Points
**Point (1, 14):**
1. For the first option, \( y = 7(1/2)^x \):
- \( y = 7(1/2)^1 = 7(1/2) = 3.5 \) (not a match)
2. For the second option, \( y = 49(2/7)^x \):
- \( y = 49(2/7)^1 = 49 \times \frac{2}{7} = 14 \) (match)
3. For the third option, \( y = 7(2/7)^x \):
- \( y = 7(2/7)^1 = 7 \times \frac{2}{7} = 2 \) (not a match)
4. For the fourth option, \( y = 49(7/2)^x \):
- \( y = 49(7/2)^1 = 49 \times \frac{7}{2} = 171.5 \) (not a match)
Only **Option 2** fits the point **(1, 14)**.
**Point (2, 4):**
Now, let’s check whether Option 2 also fits the second point (2, 4):
- For \( y = 49(2/7)^x \):
- \( y = 49(2/7)^2 = 49 \times \frac{4}{49} = 4 \) (match)
### Conclusion
Since both points (1, 14) and (2, 4) match with **Option 2** \( y = 49(2/7)^x \), this is the correct exponential equation for the graph.
**Therefore, the answer is: Option 2: \( y = 49(2/7)^x \).**
Let’s denote the exponential general form as \( y = ab^x \), where \( a \) is the initial value (when \( x=0 \)) and \( b \) is the base of the exponent.
### Evaluation of the Points
**Point (1, 14):**
1. For the first option, \( y = 7(1/2)^x \):
- \( y = 7(1/2)^1 = 7(1/2) = 3.5 \) (not a match)
2. For the second option, \( y = 49(2/7)^x \):
- \( y = 49(2/7)^1 = 49 \times \frac{2}{7} = 14 \) (match)
3. For the third option, \( y = 7(2/7)^x \):
- \( y = 7(2/7)^1 = 7 \times \frac{2}{7} = 2 \) (not a match)
4. For the fourth option, \( y = 49(7/2)^x \):
- \( y = 49(7/2)^1 = 49 \times \frac{7}{2} = 171.5 \) (not a match)
Only **Option 2** fits the point **(1, 14)**.
**Point (2, 4):**
Now, let’s check whether Option 2 also fits the second point (2, 4):
- For \( y = 49(2/7)^x \):
- \( y = 49(2/7)^2 = 49 \times \frac{4}{49} = 4 \) (match)
### Conclusion
Since both points (1, 14) and (2, 4) match with **Option 2** \( y = 49(2/7)^x \), this is the correct exponential equation for the graph.
**Therefore, the answer is: Option 2: \( y = 49(2/7)^x \).**