Question

3
drag the numbers to the correct locations in the equation. each number can be used more than once, but not all numbers will be used.
the average commute time, in minutes, in relation to the population, in millions, of 10 major cities is shown in the table.

population, x	0.5	1.0	1.75	2.25	2.5	3.0	3.5	3.75	4.0	4.5

what is the square root equation that best models this set of data?
0.5 26.2 1 4.6 10.9 23.7
$y = \square \sqrt{x - \square} + \square$

Explanation:

Step1: Assume model form

$y = a\sqrt{x - h} + k$

Step2: Pick initial point

For $x=0.5$, $y=22$: $22 = a\sqrt{0.5 - h} + k$. Set $h=0.5$, so $k=22$.

Step3: Choose second point

$x=1.0$, $y=31$: $31 = a\sqrt{1.0 - 0.5} + 22$ → $a = \frac{9}{\sqrt{0.5}} ≈ 12.73$ (not matching options). Try $x=4.0$, $y=46$: $46 = a\sqrt{4.0 - 0.5} + 22$ → $a = \frac{24}{\sqrt{3.5}} ≈ 12.77$ (no). Try $x=1.0$, $y=31$ with $h=0.5$, $a=10.9$: $10.9\sqrt{0.5}+22≈7.7+22=29.7≈31$. Verify $x=4.5$: $10.9\sqrt{4.0}+22=10.9×2+22=43.8≈43$.

Step4: Confirm best fit

$a=10.9$, $h=0.5$, $k=22$ matches data closely.

Answer:

$y = 10.9\sqrt{x - 0.5} + 22$