Find an equation in standard form of the parabola passing through the points below.
(4,-115) (3,-60) (5,-188)
To find the equation of a parabola in standard form, we need to use the general form of the equation, which is given as:
y = ax^2 + bx + c
We can substitute the x and y values of the given points into this equation to form a system of equations. Let's do that:
1) (4, -115)
-115 = 16a + 4b + c
2) (3, -60)
-60 = 9a + 3b + c
3) (5, -188)
-188 = 25a + 5b + c
Now we have a system of three equations with three unknowns (a, b, and c). We can solve this system of equations to find the values of a, b, and c.
Subtracting equation 2 from equation 1:
(-115) - (-60) = (16a + 4b + c) - (9a + 3b + c)
-55 = 7a + b
Subtracting equation 2 from equation 3:
(-188) - (-60) = (25a + 5b + c) - (9a + 3b + c)
-128 = 16a + 2b
Now we have a system of two equations with two unknowns (a and b):
7a + b = -55 --> Equation 4
16a + 2b = -128 --> Equation 5
We can solve this system of equations using any method of solving simultaneous equations. Let's solve it by eliminating b:
Multiply equation 4 by 2:
14a + 2b = -110 --> Equation 6
Subtract equation 5 from equation 6:
14a + 2b - (16a + 2b) = -110 - (-128)
-2a = 18
a = -9
Now substitute the value of a into equation 4 to solve for b:
7(-9) + b = -55
-63 + b = -55
b = -55 + 63
b = 8
Now substitute the values of a and b into any of the original equations (1, 2, or 3) to solve for c:
Using equation 1:
-115 = 16(-9) + 4(8) + c
-115 = -144 + 32 + c
-115 = -112 + c
c = -115 + 112
c = -3
The equation in standard form of the parabola passing through the given points is:
y = -9x^2 + 8x - 3