What is the equation, in standard form, of a parabola that contains the following points?

(–2, 18), (0, 2), (4, 42)

(1 point)
Responses

y = –2x2 – 2x – 3
y = –2 x 2 – 2 x – 3

y = –3x2 + 2x – 2
y = –3 x 2 + 2 x – 2

y = 3x2 – 2x + 2
y = 3 x 2 – 2 x + 2

y = –2x2 + 3x + 2

1 answer

To find the equation of a parabola in standard form, we need to use the general equation of a parabola: y = ax^2 + bx + c.

To find the values of a, b, and c, we can substitute each of the given points into the equation and form a system of three equations.

The first point (–2, 18) gives us the equation:
18 = a(-2)^2 + b(-2) + c
18 = 4a - 2b + c

The second point (0, 2) gives us the equation:
2 = a(0)^2 + b(0) + c
2 = c

The third point (4, 42) gives us the equation:
42 = a(4)^2 + b(4) + c
42 = 16a + 4b + c

Now we can solve this system of equations to find the values of a, b, and c.

From the second equation, we know that c = 2. Substituting this into the first and third equations:

18 = 4a - 2b + 2
42 = 16a + 4b + 2

Simplifying these equations:

18 - 2 = 4a - 2b
42 - 2 = 16a + 4b

16 = 4a - 2b
40 = 16a + 4b

We can solve this system of equations using elimination or substitution method.

Multiplying the first equation by 2 and adding it to the second equation:

32 = 8a - 4b
40 = 16a + 4b

72 = 24a

a = 3

Substituting a = 3 into the first equation:

16 = 4(3) - 2b
16 = 12 - 2b
-2b = 16 - 12
-2b = 4
b = -2

Now we have the values of a = 3, b = -2, and c = 2.

The equation of the parabola in standard form is:
y = 3x^2 - 2x + 2

Therefore, the correct answer is: y = 3x^2 - 2x + 2.