Cycle 2 ORP graph and Cycle 1 parabola

Key concept: This problem is about reading a sleep graph and modeling Cycle 1 with a parabola. The important skill is extracting the vertex and a second point from the diagram before writing the equation.

Step by step

(a) Equation of Cycle 1

From the graph, Cycle 1 has vertex $(2,0)$, so use

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

The point $(3.6,1.6)$ is on the curve:

$1.6=a(1.6)^2$

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

Therefore,

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

(b) First time ORP was 1.6

The axis of symmetry is $x=2$. Since $(3.6,1.6)$ is 1.6 hours to the right of the axis, the matching point is 1.6 hours to the left:

$2-1.6=0.4$

If Josh went to bed at 10 pm, then

$10:00\text{ pm} + 24\text{ minutes} = 10:24\text{ pm}$

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

Pitfall alert

A common error is mixing up ORP and OPR in the labels or reading the wrong point as the turning point. Only the vertex gives the parabola’s minimum; the matching 1.6 value comes from symmetry, not from guesswork.

Try different conditions

If the sleep graph had a different vertex, you would still use the same method: write the parabola in vertex form, substitute one other point to find $a$, and use symmetry to match equal ORP values on both sides of the axis.

Further reading

vertex form, symmetry, ORP