Question

Find the equation of the line of the best fit for the points. (-3,-40,),(1, 12),(5, 72), and 7, 137 quadratic function method

Answer 1

To find the equation of the quadratic function of the form \( y = ax^2 + bx + c \) that best fits the given points \((-3, -40)\), \((1, 12)\), \((5, 72)\), and \((7, 137)\), we can use the method of least squares.

### Step 1: Set up the equations

We have four points, which means we can set up a system of equations based on those points:

1. For \((-3, -40)\):
\[
-40 = a(-3)^2 + b(-3) + c
\]
Simplifying, we get:
\[
-40 = 9a - 3b + c \quad (1)
\]

2. For \((1, 12)\):
\[
12 = a(1)^2 + b(1) + c
\]
Simplifying, we get:
\[
12 = a + b + c \quad (2)
\]

3. For \((5, 72)\):
\[
72 = a(5)^2 + b(5) + c
\]
Simplifying, we get:
\[
72 = 25a + 5b + c \quad (3)
\]

4. For \((7, 137)\):
\[
137 = a(7)^2 + b(7) + c
\]
Simplifying, we get:
\[
137 = 49a + 7b + c \quad (4)
\]

### Step 2: Form the system of equations

Now we can write the system of equations in terms of a, b, and c:

1. \( 9a - 3b + c = -40 \) (Eq. 1)
2. \( a + b + c = 12 \) (Eq. 2)
3. \( 25a + 5b + c = 72 \) (Eq. 3)
4. \( 49a + 7b + c = 137 \) (Eq. 4)

### Step 3: Eliminate variable \(c\)

We can eliminate \(c\) from these equations by subtracting the equations as follows:

Subtract (2) from (1):
\[
(9a - 3b + c) - (a + b + c) = -40 - 12
\]
This simplifies to:
\[
8a - 4b = -52 \quad (5)
\]

Subtract (2) from (3):
\[
(25a + 5b + c) - (a + b + c) = 72 - 12
\]
This simplifies to:
\[
24a + 4b = 60 \quad (6)
\]

Subtract (3) from (4):
\[
(49a + 7b + c) - (25a + 5b + c) = 137 - 72
\]
This simplifies to:
\[
24a + 2b = 65 \quad (7)
\]

### Step 4: Solve the resulting system

Now we have a simpler system of equations:
1. \( 8a - 4b = -52 \) (5)
2. \( 24a + 4b = 60 \) (6)
3. \( 24a + 2b = 65 \) (7)

From (5):
\[
2a - b = -13 \quad (8)
\]

From (6):
\[
24a + 4b = 60 \quad (6) \implies 6a + b = 15 \quad (9)
\]
From (7):
\[
24a + 2b = 65 \quad (7) \implies 12a + b = 32.5 \quad (10)
\]

### Step 5: Solve (8) and (9):

We can solve (8) and (9) simultaneously:

From (8):
\[
b = 2a + 13 \quad (11)
\]

Substituting (11) into (9):
\[
6a + (2a + 13) = 15 \implies 8a + 13 = 15 \implies 8a = 2 \implies a = \frac{1}{4}
\]

Substituting \(a\) back to find \(b\):
\[
b = 2\left(\frac{1}{4}\right) + 13 = \frac{1}{2} + 13 = \frac{27}{2}
\]

### Step 6: Find \(c\)

Now substitute \(a\) and \(b\) back into any of the original equations to find \(c\). Using equation (2):
\[
\frac{1}{4} + \frac{27}{2} + c = 12
\]
This results in:
\[
c = 12 - \frac{1}{4} - \frac{27}{2}
\]
To perform subtraction, convert everything to a common denominator:
\[
= 12 - \frac{1}{4} - \frac{54}{4} = 12 - \frac{55}{4} = \frac{48}{4} - \frac{55}{4} = -\frac{7}{4}
\]

### Final Equation

Thus, the coefficients are:
- \(a = \frac{1}{4}\)
- \(b = \frac{27}{2}\)
- \(c = -\frac{7}{4}\)

The equation of the line of best fit is:
\[
y = \frac{1}{4}x^2 + \frac{27}{2}x - \frac{7}{4}
\]

This is the required quadratic function that fits the given points.