Question

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.

Answer 1

Answer:

1. The parabola opens up (in the shape of a "u")

2. The y intercept is a·b

3. The zeros occur at points x = a or b

Step-by-step explanation:

f(x) = y = (x - a)(x - b) = x²-a·x-b·x+a·b

y - a·b = x²-a·x-b·x = x² -x·(a + b)

y - a·b + (a + b)²/4 = x² -x·(a + b) + (a + b)²/4 = (x - (a + b)/2)²

y = (x - (a + b)/2)² + a·b - (a + b)²/4

Hence since the general form of the equation of the parabola is y = a(x - h)² + k

1. Comparing, we have a = +1 which indicates that the parabola opens up

The y intercept is found by placing x = 0 as follows;

y = (0 - (a + b)/2)² + a·b - (a + b)²/4 = a·b

The zeros is the x intercept, where y = 0 which is found as follows;

y = (0 - (a + b)/2)² + a·b - (a + b)²/4 = a·b

0 = (x - (a + b)/2)² + a·b - (a + b)²/4

Hence from the initial equation, f(x) = y = (x - a)(x - b) when y = 0, x = a or b

Hence the zeros occur at points x = a or b.