Question

Determine the quadratic regression for this data. [1, 5), (3, 4,), (4, 1), (5, 2), (7, 4) A Quadratic functions quiz answer

Answer 1

To determine a quadratic regression for the data points \((1, 5)\), \((3, 4)\), \((4, 1)\), \((5, 2)\), and \((7, 4)\), we will use the general form of a quadratic equation, which is:

\[
y = ax^2 + bx + c
\]

Where \(a\), \(b\), and \(c\) are coefficients that we need to find.

### Step 1: Set up the system of equations

We start by creating a system of equations based on the given data points:

1. For the point \((1, 5)\):
\[
5 = a(1)^2 + b(1) + c \implies a + b + c = 5 \quad \text{(1)}
\]

2. For the point \((3, 4)\):
\[
4 = a(3)^2 + b(3) + c \implies 9a + 3b + c = 4 \quad \text{(2)}
\]

3. For the point \((4, 1)\):
\[
1 = a(4)^2 + b(4) + c \implies 16a + 4b + c = 1 \quad \text{(3)}
\]

4. For the point \((5, 2)\):
\[
2 = a(5)^2 + b(5) + c \implies 25a + 5b + c = 2 \quad \text{(4)}
\]

5. For the point \((7, 4)\):
\[
4 = a(7)^2 + b(7) + c \implies 49a + 7b + c = 4 \quad \text{(5)}
\]

### Step 2: Solve the system of equations

We'll start solving with equations (1), (2), and (3):

From (1):
\[
c = 5 - a - b
\]

Substituting \(c\) into (2):
\[
9a + 3b + (5 - a - b) = 4
\]
\[
8a + 2b + 5 = 4 \implies 8a + 2b = -1 \implies 4a + b = -\frac{1}{2} \quad \text{(6)}
\]

Substituting \(c\) into (3):
\[
16a + 4b + (5 - a - b) = 1
\]
\[
15a + 3b + 5 = 1 \implies 15a + 3b = -4 \implies 5a + b = -\frac{4}{3} \quad \text{(7)}
\]

### Step 3: Eliminate \(b\)

Subtract (6) from (7):
\[
(5a + b) - (4a + b) = -\frac{4}{3} + \frac{1}{2}
\]
\[
a = -\frac{4}{3} + \frac{3}{6} = -\frac{8}{6} + \frac{3}{6} = -\frac{5}{6}
\]

### Step 4: Find \(b\) and \(c\)
Substituting \(a\) back into (6):
\[
4(-\frac{5}{6}) + b = -\frac{1}{2} \implies -\frac{20}{6} + b = -\frac{1}{2}
\]
\[
b = -\frac{1}{2} + \frac{10}{3} = -\frac{3}{6} + \frac{20}{6} = \frac{17}{6}
\]

Substituting \(a\) and \(b\) back into (1) to find \(c\):
\[
-\frac{5}{6} + \frac{17}{6} + c = 5
\]
\[
c = 5 - \frac{12}{6} = 5 - 2 = 3
\]

### Step 5: Final result
Now we have the coefficients:
\[
a = -\frac{5}{6}, \quad b = \frac{17}{6}, \quad c = 3
\]

Thus, the quadratic regression function is:

\[
y = -\frac{5}{6}x^2 + \frac{17}{6}x + 3
\]

This represents the best fitting quadratic curve for the given data points.