What is the equation in the standard form of a parabola that models the values in the table x -1, 0, 2 f(x) 12, 5, 15?

Solution:

The data given can be summarised in the table given below:

x	f(x)
-1	12
0	5
2	15

Parabola takes the form of the equation

y = f(x) = ax² + bx + c and a ≠ 0 --- (1)

The objective here is to find out the values of a, b and c.

This can be done with the help of the above table where values of x and corresponding values of f(x)/y are given:

The value of c is obtained when x = 0.

From the above table we have y = 5 when x = 0.

The equation (1) gives us the value of c as:

5 = a(0)² + b(0) + c

c = 5 --- (2)

From the other point (-1, 12) we get the following:

12 = a(-1)² + b(-1) + 5

⇒ a - b = 12 - 5 = 7 --- (3)

From the third point (2, 15) we get

15 = a(2)² + b(2) + 5

4a + 2b = 10

⇒ 2a + b = 5 --- (4)

Solving (3) and (4) simultaneously we get

a = 4 and b = -3

Therefore the equation of the parabola is

y = f(x) = 4x² - 3x + 5

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What is the equation in the standard form of a parabola that models the values in the table

Summary:

The equation of the parabola obtained is y = f(x) = 4x² - 3x + 5.