Step-by-step explanation:
The second part of the roller coaster must be a parabola with the expression
(x-a)(x-b)
let a= 1 and b= 2
so the equation is :
f(x)= (x-1)(x-2) = x²-2x-x+2 = x²-3x+2
- The vertex of the parabola:
f(x) = x²-3x+2
The vertex coordinates are ( , f(
-b/2a = 3/1*2 = -3/2
f(-b/2a)= 8.75 (using a calculator)
a is positive since 1 > 0
so :
- f decreases untill reaching (-3/2 , 8.75) and then increases
- the vertex (-3/2, 8.75) is a minimum since f" > 0
f(x) = x²-3x+2
f'(x)= 2x-3
f"(x)= 2 > 0
no need to derivate you can just notice that a is positive
the y-intercept is given by f(0)
f(0) = 0²-3*0+2 = 2
so the y-intercept (0,2)
the x-intercept is given by f(x)= 0
x²-3x+2 = 0
The dicriminant method:
let Δ be our dicriminant
- Δ = b²-4*a*c
- Δ= -3²-4*1*2
- Δ= 9-8
- Δ= 1
Δ > 0 so we have two solutions x and x'
x= (-b-√Δ)/2a = (3-1)/2 = 2/2 = 1
x'=(-b+√Δ)/2a = (3+1)/2 = 4/2 = 2
the x-intercepts are : (1,0) (2,0)