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
<u>y-intercept</u>
when x = 0
⇒ f(0) = 5(0²) - 40(0) + 75 = 75
⇒ y-intercept = (0, 75)
<u>x-intercepts</u>
x = 3 and x = 5
<u>Vertex</u>
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)
