Odds Ratio Product (ORP) Cycle 1 parabola and line of symmetry

Key concept: This question combines a parabola model with symmetry. First identify the vertex and a point on the graph, then use the axis of symmetry to match equal ORP values at equal distances from the vertex.

Step by step

(a) Equation of Cycle 1

From the graph, Cycle 1 is fully asleep at $(2,0)$, so the vertex is

$(2,0)$

Use vertex form:

$y=a(x-2)^2$

The point $(3.6,1.6)$ lies on the parabola, so substitute it in:

$1.6=a(3.6-2)^2$

$1.6=a(1.6)^2=a(2.56)$

$a=\frac{1.6}{2.56}=0.625=\frac{5}{8}$

So the equation is

$\boxed{y=\frac{5}{8}(x-2)^2}$

(b) First time ORP was 1.6

The line of symmetry is

$x=2$

The point $(3.6,1.6)$ is $3.6-2=1.6$ hours to the right of the axis. The matching point with the same ORP is the same distance to the left:

$2-1.6=0.4$

So the watch first recorded ORP $1.6$ at

$10\text{ pm} + 0.4\text{ h} = 10:24\text{ pm}$

$\boxed{10:24\text{ pm}}$

Pitfall alert

Do not use the point $(3.6,1.6)$ as the vertex. The vertex is the minimum point $(2,0)$, and the symmetry calculation must be done from the axis $x=2$, not from the y-value. Also, be careful converting $0.4$ hours into minutes: $0.4\times 60=24$ minutes.

Try different conditions

If the vertex were shifted to $(h,k)$, the model would become $y=a(x-h)^2+k$. The symmetry method would still work the same way: any ORP value on one side of the axis has a matching point the same horizontal distance from $x=h$ on the other side.

Further reading

vertex form, axis of symmetry, parabola