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

x = -2 0 4

f(x) = 4 -6 70

a. 3x^2 + 4x - 6
b. 4x^2 + 3x - 6
c. 4x^2 - 3x - 6
d. -4x^2 - 3x - 6

  b.  4x^2 + 3x - 6

Step-by-step explanation:

The values of f(x) for the extremes of x are more positive than the value of f(x) for the middle x, so we know the parabola opens upward. That eliminates choice D.

It is probably easiest to evaluate the other expressions to see which one matches the given f(x) values. For the purpose, it is usually easier to use the Horner form of the equation.

a. f(-2) = (3(-2) +4)(-2) -6 = -2(-2) -6 = -2 ≠ 4

b. f(-2) = (4(-2) +3)(-2) -6 = -5(-2) -6 = 4 . . . . matches the given data point

c. Because (b) matches, we know this one will not.

The appropriate choice is B.

b. 4x² -3x - 6  

Step-by-step explanation:

This is a multiple-choice question, so the easiest way to get the correct answer would be to insert the values and see which option works.

However, "they" may not always give you the answer. Here's how to work it out.

The equation for a parabola is

y = ax² + bx + g

You can substitute the points (x, y) into the equation three times and then solve three equations in three unknowns.

Point (-2, 4)

     a(-2)² + b(-2) + c = 4

(1) 4a      - 2b      + c = 4

Point (0, -6)

   a(0)² + b(0) + c = -6

(2)                      c = -6

Point (4, 70)

       a(4)² +   b(4) + c = 70

(3) 16a      + 4b     + c = 70

From equation (2),  

c = -6

Substitute the value of c into equations (1) and (3).

(4)  4a  - 2b - 6 =   4

(5) 16a + 4b - 6 = 70

Move the constants to the right-hand side

(6)  4a - 2b  = 10

(7) 16a + 4b = 76

Divide equation (7) by 2

(8) 4a  - 2b  = 10

(9) 8a + 2b  = 38

Add equations (8) and (9)

12a = 48

   a = 48/12 = 4

Substitute the values of  a and c into equation (1)

4(4) - 2b  -  6 =  4

  16 - 2b  -  6 =  4

        -2b + 10 =  4

        -2b        = -6

            b        =  3

So, a = 4, b = 3, c = -6

The equation for the parabola is

y = 4x² + 3x - 6

The figure below shows your parabola passing through the three points.