Question

Given points (1,0) and (2,0) the axis is x = 1.5. Using (3,1) shows the parabola opens upward, so it decreases to the vertex at x = 1.5 then increases after.
Responces are. 0, 1, 2, 3, 4, -infinity, infinity

Answer 1

Take the quadratic with roots 1 and 2 and passing through (3,1):
f(x) = a(x−1)(x−2). Using f(3)=1 gives 2a = 1 ⇒ a = 1/2, so
f(x) = (1/2)(x−1)(x−2) = 1/2 x^2 − 3/2 x + 1.

Vertex: x = 1.5, f(1.5) = −1/8. Since a = 1/2 > 0 the parabola opens upward.
- Decreasing on (−∞, 1.5)
- Increasing on (1.5, ∞)
- Domain: (−∞, ∞)
- Range: [−1/8, ∞)
- Zeros: x = 1 and x = 2
- Sample values: f(0)=1, f(1)=0, f(2)=0, f(3)=1, f(4)=3

So the line x = 1.5 is the axis; it decreases until 1.5 then increases afterwards; domain uses −∞ and ∞; the listed integers 0,1,2,3,4 appear as x-values or function values as shown above.