Question

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

Answer 1

To find the equation of a parabola that contains three points, we can use the standard form equation of a parabola, which is:

y = ax^2 + bx + c

We can substitute the x and y values of each point into this equation to form three simultaneous equations. Using the given points (-2, 18), (0, 2), and (4, 42), we get:

18 = 4a - 2b + c ...(1)
2 = 0a + 0b + c ...(2)
42 = 16a + 4b + c ...(3)

We can solve these simultaneous equations to find the values of a, b, and c. Subtracting equation (2) from equation (1), we get:

16 = 4a - 2b ...(4)

Subtracting equation (2) from equation (3), we get:

40 = 16a + 4b ...(5)

Simplifying equation (4) and (5):

8 = 2a - b ...(6)
20 = 8a + 2b ...(7)

Multiplying equation (6) by 2 and adding it to equation (7):

16 + 20 = 4a + a
36 = 5a
a = 36/5 = 7.2

Substituting the value of a back into equation (6), we get:

8 = 2(7.2) - b
8 = 14.4 - b
b = 14.4 - 8
b = 6.4

Now we can substitute the values of a and b into equation (2) to find the value of c:

2 = 0(7.2)^2 + 6.4(7.2) + c
2 = 46.08 + c
c = 2 - 46.08
c = -44.08

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

y = 7.2x^2 + 6.4x - 44.08