How can we find a quadratic model for the data set consisting of the values (2, 20), (0, 4), and (4, 20)?

To find a quadratic model for the given set of values, we start by denoting the data points as follows:

• (0, 4)
• (2, 20)
• (4, 20)

We want to find a quadratic equation of the form:

y = ax² + bx + c

where a, b, and c are constants we need to determine using the provided points.

Step 1: Set up the equations

Substituting each point into the equation:

1. For (0, 4):

4 = a(0)² + b(0) + c → c = 4

2. For (2, 20):

20 = a(2)² + b(2) + c → 20 = 4a + 2b + 4

Which simplifies to:

4a + 2b = 16 → 2a + b = 8 (Equation 1)

3. For (4, 20):

20 = a(4)² + b(4) + c → 20 = 16a + 4b + 4

Which simplifies to:

16a + 4b = 16 → 4a + b = 4 (Equation 2)

Step 2: Solve the equations

We now have two equations:

• Equation 1: 2a + b = 8
• Equation 2: 4a + b = 4

To solve for a and b, we can subtract Equation 1 from Equation 2:

(4a + b) – (2a + b) = 4 – 8

which simplifies to:

2a = -4 → a = -2

Substituting a back into Equation 1 to find b:

2(-2) + b = 8

-4 + b = 8 → b = 12

Final Equation

Now that we have the coefficients:

• a = -2
• b = 12
• c = 4

The quadratic model that represents the set of values is:

y = -2x² + 12x + 4

Conclusion

This quadratic function should fit the data points (0, 4), (2, 20), and (4, 20) appropriately.