To find a quadratic model for the given values: (2, 20), (0, 4), (4, 4), and (20), we will use the standard form of a quadratic equation, which is:
y = ax² + bx + c
Here, (x, y) represents the points we have. Our goal is to determine the coefficients a, b, and c.
First, we will arrange our known points as follows:
- (0, 4)
- (2, 20)
- (4, 4)
- (20, ?)
From the point (0,4), we substitute x = 0 in the quadratic equation to find c:
y = a(0)² + b(0) + c
Thus, we get:
c = 4
Now we can rewrite our equation as:
y = ax² + bx + 4
Next, we can use points (2, 20) and (4, 4) to create a system of equations. Substituting x = 2 and x = 4:
For (2, 20):
20 = a(2)² + b(2) + 4
20 = 4a + 2b + 4
Rearranging it gives:
4a + 2b = 16
or
2a + b = 8 (Equation 1)
For (4, 4):
4 = a(4)² + b(4) + 4
4 = 16a + 4b + 4
Rearranging it gives:
16a + 4b = 0
or
4a + b = 0 (Equation 2)
Now we have a system of two equations:
- Equation 1: 2a + b = 8
- Equation 2: 4a + b = 0
We can solve for b in terms of a using Equation 2:
b = -4a
Now, substitute b into Equation 1:
2a – 4a = 8
-2a = 8
Thus, we find:
a = -4
Now, substituting a back into the equation for b:
b = -4(-4) = 16
So, we have:
- a = -4
- b = 16
- c = 4
Putting it all together, the quadratic model is:
y = -4x² + 16x + 4
This equation represents the quadratic model for the given set of values.