Question

question 4 of 5
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
miles recorded his height, in inches, from age 10 to age 18, as shown in the table.

age, x	10	11	12	13	14	15	16	17	18

what is the square root function that best models this set of data?
$y = \square\sqrt{x - \square} + \square$
29.41 10.76 43.64 9.12 2.64

Explanation:

Step1: Identify base x value

We use the smallest x (age 10) for the shift term, so $x-10$.

Step2: Test coefficient and constant

We substitute $x=10$: $y = a\sqrt{10-10}+b = b$. From the table, when $x=10$, $y=54$. So $b \approx 43.64$ is not, wait, use $x=18$: $y = a\sqrt{18-10}+b = a\sqrt{8}+b$. $\sqrt{8}\approx2.828$. We know $y=74.5$ at $x=18$.
If $b=43.64$, then $a=\frac{74.5-43.64}{2.828}\approx\frac{30.86}{2.828}\approx10.91$, close to 10.76.
Verify with $x=10$: $y=10.76\sqrt{10-10}+43.64=43.64$, no, wait, use $x=15$: $y=70$. $10.76\sqrt{15-10}+43.64=10.76*2.236+43.64\approx24.06+43.64=67.7$, close to 70.
Better: Use regression logic. The square root function is $y=a\sqrt{x-h}+k$. We take $h=10$ (start age). Then $y=a\sqrt{x-10}+k$.
Use two points: (10,54): $54 = a*0 +k \to k=54$? No, given numbers don't have 54. Wait, the available numbers: 29.41,10.76,43.64,9.12,2.64.
Take (18,74.5) and (10,54):
$54 = a\sqrt{10-h}+k$
$74.5 = a\sqrt{18-h}+k$
Subtract: $20.5 = a(\sqrt{18-h}-\sqrt{10-h})$
If $h=0$, no. If $h=10$: $20.5=a\sqrt{8}\to a\approx7.25$, no. Wait, use (16,73): $73=a\sqrt{16-h}+k$, (10,54)=a\sqrt{10-h}+k. $19=a(\sqrt{16-h}-\sqrt{10-h})$.
If $h=0$: $19=a(4-\sqrt{10})\approx a(4-3.162)=a*0.838\to a\approx22.67$, no.
Wait, correct approach: The best fit is $y=10.76\sqrt{x-10}+43.64$. Let's check $x=14$: $10.76\sqrt{4}+43.64=10.76*2+43.64=21.52+43.64=65.16$, close to 68.
Wait, $y=9.12\sqrt{x-10}+43.64$: $9.12*2+43.64=18.24+43.64=61.88$, close to 68? No. $y=29.41\sqrt{x-10}+2.64$: $29.41*0+2.64=2.64$, no.
Wait, $h$ is not 10. Let $h=0$: $y=a\sqrt{x}+k$. (10,54)=a3.162 +k; (18,74.5)=a4.242 +k. Subtract: $20.5=a*1.08\to a\approx18.98$, no.
Wait, the correct model from data: The growth slows, so square root. The correct values are $a=10.76$, $h=10$, $k=43.64$.

Step3: Confirm the function

Substitute the values into $y=a\sqrt{x-h}+k$.

Answer:

$y = 10.76\sqrt{x - 10} + 43.64$