IB Mathematics SL 5.6 Local maximum and minimum points AI SL Paper 1- Exam Style Questions- New Syllabus

(a)
The axis of symmetry lies midway between the two equal‑height points:
\( d = \frac{3 + 8.5}{2} = 5.75 \)
 \(\boxed{d = 5.75}\)

(b)
Using the axis‑of‑symmetry formula \(d = -\frac{b}{2a}\) with \(a = -0.2\):
\( 5.75 = -\frac{b}{2(-0.2)} \) → \( 5.75 = \frac{b}{0.4} \) → \( b = 2.3 \)
\(\boxed{2.3}\)

(c)
The dolphin re‑enters the water when \(h(d) = 0\):
\( -0.2d^{2} + 2.3d = 0 \) → \( d(-0.2d + 2.3) = 0 \)
Thus \(d = 0\) (start) or \(d = \frac{2.3}{0.2} = 11.5\).
\(\boxed{11.5 \ \text{m}}\)

(d)
A negative \(h\)-value would mean the dolphin is below the water surface (underwater). In the context of the model, however, the function is only intended to describe the jump above the water, so negative values are not physically meaningful for this model.