What is the equation in standard form of a parabola that contains the following points 2, 18; 0, 4; 4, 24?

To find the equation of a parabola that passes through the points (2, 18), (0, 4), and (4, 24), we start with the standard form of a parabola:

y = ax² + bx + c

We need to determine the coefficients a, b, and c. We can do this by substituting the points into the equation to create a system of equations.

1. For the point (2, 18):

18 = a(2²) + b(2) + c
18 = 4a + 2b + c

2. For the point (0, 4):

4 = a(0²) + b(0) + c
4 = c

3. For the point (4, 24):

24 = a(4²) + b(4) + c
24 = 16a + 4b + c

Now we substitute c = 4 into the first and third equations:

1. 18 = 4a + 2b + 4
14 = 4a + 2b
7 = 2a + b (Equation 1)

2. 24 = 16a + 4b + 4
20 = 16a + 4b
5 = 4a + b (Equation 2)

Now we have a system of two equations:

Equation 1: 7 = 2a + b

Equation 2: 5 = 4a + b

Next, we can eliminate b by subtracting Equation 1 from Equation 2:

(5 = 4a + b) – (7 = 2a + b)
5 – 7 = 4a – 2a
-2 = 2a
a = -1

Now we substitute a back into Equation 1 to find b:

7 = 2(-1) + b
7 = -2 + b
b = 9

Now we have a = -1, b = 9, and c = 4. Therefore, the equation of the parabola in standard form is:

y = -x² + 9x + 4