Question

A roller coaster at XTRA Fun Rides goes into an underground tunnel during the first descent of the

Answer 1

Answer:

y-intercept:  (0, 75) ⇒ y = 75

x-intercepts:  (3, 0) and (5, 0) ⇒ x = 3 and x = 5

vertex: (4, -5)

Step-by-step explanation:

Let the x-axis be the ground (when y = 0).  Therefore underground when y < 0 and overground when y > 0

The function is a quadratic:  f(x) = ax² + bx + c

If the coaster enters the tunnel three seconds after the first descent starts and stays in the tunnel for a total of 2 seconds, then y = 0 when x = 3 and x = 5

Also, when x = 1, y = 40

• f(3) = 0
  ⇒ 9a + 3b + c = 0

• f(5) = 0
  ⇒ 25a + 5b + c = 0

• f(1) = 40
  ⇒ a + b + c = 40

Solve 2 equations simultaneously:

• 9a + 3b + c = 0
• 25a + 5b + c = 0
• a + b + c = 40

⇒ a = 5  b = -40  c = 75

⇒ f(x) = 5x² - 40x + 75

y-intercept

when x = 0

⇒ f(0) = 5(0²) - 40(0) + 75 = 75

⇒ y-intercept = (0, 75)

x-intercepts

x = 3 and x = 5

Vertex

If the x-intercepts are when x = 3 and x = 5, then vertex is when x = 4

⇒ f(4) = 5(4²) - 40(4) + 75 = -5

⇒ vertex = (4, -5)