Question
Use the table to answer the question. x y 62 104 64 34 67 4 69 29 71 98 Write the quadratic function for the given set of data. What is the value of b? Round the value to the nearest whole number. (1 point) In the quadratic function that represents the dataset, b =
Answers
GPT-4o mini
To find the quadratic function that best fits the given data points, we'll first consider a general form of a quadratic function:
\[
y = ax^2 + bx + c
\]
We have the following data points from the table:
1. (62, 104)
2. (64, 34)
3. (67, 4)
4. (69, 29)
5. (71, 98)
We need to use these points to determine the coefficients \(a\), \(b\), and \(c\).
### Steps to determine \(a\), \(b\), and \(c\):
1. **Set up the system of equations.** We can use three points to create a system of equations. For example, using the first three points:
- For \( (62, 104) \):
\[
104 = a(62)^2 + b(62) + c
\]
- For \( (64, 34) \):
\[
34 = a(64)^2 + b(64) + c
\]
- For \( (67, 4) \):
\[
4 = a(67)^2 + b(67) + c
\]
2. **Simplify the equations:**
- Equation 1:
\[
104 = 3844a + 62b + c
\]
- Equation 2:
\[
34 = 4096a + 64b + c
\]
- Equation 3:
\[
4 = 4489a + 67b + c
\]
3. **Eliminate \(c\)** by subtracting pairs of equations. For example, subtract equation 1 from equation 2:
\[
(34 - 104) = (4096 - 3844)a + (64 - 62)b + (c-c)
\]
This simplifies to:
\[
-70 = 252a + 2b \quad \text{(Equation A)}
\]
Now subtract equation 2 from equation 3:
\[
(4 - 34) = (4489 - 4096)a + (67 - 64)b + (c - c)
\]
This simplifies to:
\[
-30 = 393a + 3b \quad \text{(Equation B)}
\]
4. **Now you have a system of two equations (A and B):**
- Equation A: \(-70 = 252a + 2b\)
- Equation B: \(-30 = 393a + 3b\)
5. **Solve this system for \(a\) and \(b\).**
Multiply Equation A by 3:
\[
-210 = 756a + 6b \quad \text{(Equation A1)}
\]
Multiply Equation B by 2:
\[
-60 = 786a + 6b \quad \text{(Equation B1)}
\]
Now subtract A1 from B1:
\[
(-60 + 210) = (786 - 756)a + (6b - 6b)
\]
This simplifies to:
\[
150 = 30a \implies a = 5
\]
6. **Substitute \(a = 5\) into one of the original equations to find \(b\). Use Equation A:**
\[
-70 = 252(5) + 2b
\]
\[
-70 = 1260 + 2b
\]
\[
2b = -70 - 1260 = -1330
\]
\[
b = -665
\]
### Conclusion
Thus, the value of \(b\) is \(-665\).
If you meant to ask for \(b\) rounded to the nearest whole number, it is \(-665\). Please let me know if you would like to find \(c\) or need any additional help with this problem!
\[
y = ax^2 + bx + c
\]
We have the following data points from the table:
1. (62, 104)
2. (64, 34)
3. (67, 4)
4. (69, 29)
5. (71, 98)
We need to use these points to determine the coefficients \(a\), \(b\), and \(c\).
### Steps to determine \(a\), \(b\), and \(c\):
1. **Set up the system of equations.** We can use three points to create a system of equations. For example, using the first three points:
- For \( (62, 104) \):
\[
104 = a(62)^2 + b(62) + c
\]
- For \( (64, 34) \):
\[
34 = a(64)^2 + b(64) + c
\]
- For \( (67, 4) \):
\[
4 = a(67)^2 + b(67) + c
\]
2. **Simplify the equations:**
- Equation 1:
\[
104 = 3844a + 62b + c
\]
- Equation 2:
\[
34 = 4096a + 64b + c
\]
- Equation 3:
\[
4 = 4489a + 67b + c
\]
3. **Eliminate \(c\)** by subtracting pairs of equations. For example, subtract equation 1 from equation 2:
\[
(34 - 104) = (4096 - 3844)a + (64 - 62)b + (c-c)
\]
This simplifies to:
\[
-70 = 252a + 2b \quad \text{(Equation A)}
\]
Now subtract equation 2 from equation 3:
\[
(4 - 34) = (4489 - 4096)a + (67 - 64)b + (c - c)
\]
This simplifies to:
\[
-30 = 393a + 3b \quad \text{(Equation B)}
\]
4. **Now you have a system of two equations (A and B):**
- Equation A: \(-70 = 252a + 2b\)
- Equation B: \(-30 = 393a + 3b\)
5. **Solve this system for \(a\) and \(b\).**
Multiply Equation A by 3:
\[
-210 = 756a + 6b \quad \text{(Equation A1)}
\]
Multiply Equation B by 2:
\[
-60 = 786a + 6b \quad \text{(Equation B1)}
\]
Now subtract A1 from B1:
\[
(-60 + 210) = (786 - 756)a + (6b - 6b)
\]
This simplifies to:
\[
150 = 30a \implies a = 5
\]
6. **Substitute \(a = 5\) into one of the original equations to find \(b\). Use Equation A:**
\[
-70 = 252(5) + 2b
\]
\[
-70 = 1260 + 2b
\]
\[
2b = -70 - 1260 = -1330
\]
\[
b = -665
\]
### Conclusion
Thus, the value of \(b\) is \(-665\).
If you meant to ask for \(b\) rounded to the nearest whole number, it is \(-665\). Please let me know if you would like to find \(c\) or need any additional help with this problem!