Question

WILL MARK BRAINLIEST!!! PLZ HELP!!! 25 POINTS!! Ray needs help creating the second part of the coaster. Create a unique parabola in the pattern f(x) = (x − a)(x − b). Describe the direction of the parabola and determine the y-intercept and zeros then create a graph of the polynomial function

Answer 1

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

• a= 1
• b= -3
• c= 2

The vertex coordinates are ($\frac{-b}{2a}$ , 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

the y-intercept is given by f(0)

f(0) = 0²-3*0+2 = 2

so the y-intercept (0,2)

• the x-intercept

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)

• here is a graph: