What is the equation in the standard form of a parabola that contains the following points: (2, 18), (0, 2), (4, 42)?

To find the equation of a parabola in standard form, which is typically expressed as y = ax^2 + bx + c, we can use the given points to create a system of equations.

We have three points: (2, 18), (0, 2), and (4, 42). We can substitute these points into the standard form equation to form three equations.

1. For the point (2, 18):
18 = a(2)^2 + b(2) + c
This simplifies to 18 = 4a + 2b + c.

2. For the point (0, 2):
2 = a(0)^2 + b(0) + c
This simplifies to 2 = c.

3. For the point (4, 42):
42 = a(4)^2 + b(4) + c
This simplifies to 42 = 16a + 4b + c.

Now, we can substitute c = 2 into the first and third equations:

1. 18 = 4a + 2b + 2
   Subtract 2 from both sides:
   16 = 4a + 2b, which simplifies to 8 = 2a + b or b = 8 - 2a.
2. 42 = 16a + 4b + 2
   Subtract 2 from both sides:
   40 = 16a + 4b. Now we can substitute b = 8 - 2a:
   40 = 16a + 4(8 - 2a)
   This simplifies to 40 = 16a + 32 - 8a, leading to 40 = 8a + 32, thus 8 = 8a, or a = 1.

With a = 1 and c = 2, substitute back to find b:
b = 8 - 2(1) = 6.

So the values are:
a = 1, b = 6, c = 2.

Therefore, the equation of the parabola in standard form is:

y = 1x^2 + 6x + 2 or simply y = x^2 + 6x + 2.