Question

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

ride. The coaster enters the tunnel three seconds after the first descent starts and stays in the
tunnel for a total of 2 seconds. The roller coaster Is 40 feet above ground
one second after the
Initial drop. The graph below represents the height of the roller coaster versus the time passed
since the initial drop.

Answer 1

y-intercept: (0, 75) ⇒ y = 75

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

vertex: (4, -5)

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)