Question
Find the quadratic function that models the data in the table below.
x
negative 2
negative 1
0
1
2
3
4
5
6
7
8
9
10
y
18
7
0
negative 3
negative 2
3
12
25
42
63
88
117
150
Question content area bottom
Part 1
The equation of the quadratic function that models the given data is yequalsnothing x squared plus left parenthesis nothing right parenthesis x plus nothing
.
x
negative 2
negative 1
0
1
2
3
4
5
6
7
8
9
10
y
18
7
0
negative 3
negative 2
3
12
25
42
63
88
117
150
Question content area bottom
Part 1
The equation of the quadratic function that models the given data is yequalsnothing x squared plus left parenthesis nothing right parenthesis x plus nothing
.
Answers
GPT-4o mini
To find the quadratic function that models the given data, we need to determine a function of the form:
\[
y = ax^2 + bx + c
\]
where \( a \), \( b \), and \( c \) are constants that we need to find.
We can use the data in the table to set up a system of equations. We'll pick three points from the data, specifically \((-2, 18)\), \((0, 0)\), and \((2, 3)\).
**Using the points:**
1. For \( x = -2 \), \( y = 18 \):
\[
18 = a(-2)^2 + b(-2) + c
\]
\[
18 = 4a - 2b + c \quad (1)
\]
2. For \( x = 0 \), \( y = 0 \):
\[
0 = a(0)^2 + b(0) + c
\]
\[
0 = c \quad (2)
\]
3. For \( x = 2 \), \( y = 3 \):
\[
3 = a(2)^2 + b(2) + c
\]
\[
3 = 4a + 2b + c \quad (3)
\]
**Using equation \( (2) \)**, we substitute \( c = 0 \) into equations \( (1) \) and \( (3) \):
1. Substituting \( c = 0 \) into equation \( (1) \):
\[
18 = 4a - 2b \quad (4)
\]
2. Substituting \( c = 0 \) into equation \( (3) \):
\[
3 = 4a + 2b \quad (5)
\]
**Now we can solve equations \( (4) \) and \( (5) \):**
From equation \( (4) \):
\[
4a - 2b = 18 \implies 2a - b = 9 \quad (6)
\]
From equation \( (5) \):
\[
4a + 2b = 3 \implies 2a + b = \frac{3}{2} \quad (7)
\]
Now, we can add equations \( (6) \) and \( (7) \):
\[
(2a - b) + (2a + b) = 9 + \frac{3}{2}
\]
\[
4a = 9 + 1.5 = 10.5 \implies a = \frac{10.5}{4} = 2.625
\]
Now substituting \( a \) back into equation \( (6) \) to find \( b \):
\[
2(2.625) - b = 9
\]
\[
5.25 - b = 9 \implies b = 5.25 - 9 = -3.75
\]
**Now we have:**
- \( a \approx 2.625 \)
- \( b \approx -3.75 \)
- \( c = 0 \)
Finally, the quadratic function is approximately:
\[
y = 2.625x^2 - 3.75x
\]
**Now rounding to the nearest whole numbers:**
Typically, for many practical purposes, we represent the quadratic function as:
\[
y = \frac{21}{8}x^2 - \frac{15}{4}x \quad \text{(exact form)}
\]
The final answer in decimal approximation is:
\[
\text{The equation of the quadratic function} \quad y = 2.625 x^2 - 3.75 x
\]
So, the full answer in function form would be:
\[
y = 2.625x^2 - 3.75x + 0
\]
\[
y = ax^2 + bx + c
\]
where \( a \), \( b \), and \( c \) are constants that we need to find.
We can use the data in the table to set up a system of equations. We'll pick three points from the data, specifically \((-2, 18)\), \((0, 0)\), and \((2, 3)\).
**Using the points:**
1. For \( x = -2 \), \( y = 18 \):
\[
18 = a(-2)^2 + b(-2) + c
\]
\[
18 = 4a - 2b + c \quad (1)
\]
2. For \( x = 0 \), \( y = 0 \):
\[
0 = a(0)^2 + b(0) + c
\]
\[
0 = c \quad (2)
\]
3. For \( x = 2 \), \( y = 3 \):
\[
3 = a(2)^2 + b(2) + c
\]
\[
3 = 4a + 2b + c \quad (3)
\]
**Using equation \( (2) \)**, we substitute \( c = 0 \) into equations \( (1) \) and \( (3) \):
1. Substituting \( c = 0 \) into equation \( (1) \):
\[
18 = 4a - 2b \quad (4)
\]
2. Substituting \( c = 0 \) into equation \( (3) \):
\[
3 = 4a + 2b \quad (5)
\]
**Now we can solve equations \( (4) \) and \( (5) \):**
From equation \( (4) \):
\[
4a - 2b = 18 \implies 2a - b = 9 \quad (6)
\]
From equation \( (5) \):
\[
4a + 2b = 3 \implies 2a + b = \frac{3}{2} \quad (7)
\]
Now, we can add equations \( (6) \) and \( (7) \):
\[
(2a - b) + (2a + b) = 9 + \frac{3}{2}
\]
\[
4a = 9 + 1.5 = 10.5 \implies a = \frac{10.5}{4} = 2.625
\]
Now substituting \( a \) back into equation \( (6) \) to find \( b \):
\[
2(2.625) - b = 9
\]
\[
5.25 - b = 9 \implies b = 5.25 - 9 = -3.75
\]
**Now we have:**
- \( a \approx 2.625 \)
- \( b \approx -3.75 \)
- \( c = 0 \)
Finally, the quadratic function is approximately:
\[
y = 2.625x^2 - 3.75x
\]
**Now rounding to the nearest whole numbers:**
Typically, for many practical purposes, we represent the quadratic function as:
\[
y = \frac{21}{8}x^2 - \frac{15}{4}x \quad \text{(exact form)}
\]
The final answer in decimal approximation is:
\[
\text{The equation of the quadratic function} \quad y = 2.625 x^2 - 3.75 x
\]
So, the full answer in function form would be:
\[
y = 2.625x^2 - 3.75x + 0
\]