Find an equation in standard form of the parabola passing through the points below.
(1,8) (2,3) (4,-31)
To find the equation of a parabola passing through these points, we can use the standard form equation of a parabola: y = ax^2 + bx + c.
Substituting the x and y values of the first point (1,8), we get:
8 = a(1)^2 + b(1) + c
8 = a + b + c
Substituting the x and y values of the second point (2,3), we get:
3 = a(2)^2 + b(2) + c
3 = 4a + 2b + c
Substituting the x and y values of the third point (4,-31), we get:
-31 = a(4)^2 + b(4) + c
-31 = 16a + 4b + c
Now, we have a system of three equations with three unknowns (a, b, c). Let's solve it.
Rearranging the first equation:
a + b + c = 8 --> c = 8 - a - b
Substituting this value of c in the second equation:
3 = 4a + 2b + (8 - a - b)
3 = 3a + b + 8
Rearranging this equation:
b = -3a + (-5)
Substituting these values of a and b in the third equation:
-31 = 16a + 4(-3a - 5) + (8 - a - (-3a - 5))
-31 = 16a - 12a - 20 + 8 -a + 3a + 5
-31 = 6a - 7
Rearranging this equation:
6a = -31 + 7
6a = -24
a = -4
Now, substituting the value of a in the equation b = -3a - 5:
b = -3(-4) - 5
b = 12 - 5
b = 7
Finally, substituting the values of a and b in c = 8 - a - b:
c = 8 - (-4) - 7
c = 8 + 4 - 7
c = 5
Therefore, the equation in standard form of the parabola passing through the points (1,8), (2,3), and (4,-31) is:
y = -4x^2 + 7x + 5