Question 11·SAT Desmos Regression Practice·Advanced Math

In $E(2)=91$, the filter has been used for $2$ hours and its efficiency is $91\%$, so the point is $(2,91)$. Enter the other two conditions the same way: hours in $x_1$, efficiencies in $y_1$. Desmos shows the paired rows $(2,91)$, $(5,82)$, and $(9,58)$.

Type $y_1\sim ax_1^2+bx_1+c$. The regression symbol $\sim$ tells Desmos to find one set of coefficients that fits all three rows. Under PARAMETERS, it reports $a\approx-0.428571$. Read $a$, the squared-term coefficient, rather than $b$ or $c$.

Type $a$ on a new line and tap that line's fraction button. Desmos shows $-\frac{3}{7}$. So the coefficient of $x^2$ in the filtration-efficiency model is $-\frac{3}{7}$. Choice D.

An average change per hour is the change in efficiency divided by the hours elapsed. Type $\frac{82-91}{5-2}$; Desmos shows $-3$ percentage points per hour. The denominator is the 3-hour gap, not the ending hour $5$.

For the next pair, type $\frac{58-82}{9-5}$. Desmos shows $-6$ percentage points per hour. Compare these per-hour rates, not the raw efficiency drops: the two time gaps have different lengths.

For hours $u$ and $v$, subtract the model's outputs so $c$ cancels:

Use the difference of squares, $v^2-u^2=(v-u)(v+u)$, and factor out $v-u$:

Divide by the hours elapsed:

So the rate depends on the sum of the two hours.

Apply that rate rule to each pair of hours:

Both equations contain the same $b$, so subtracting them will remove it.

Subtract the first equation from the second, which cancels $b$:

Divide by the bracketed difference:

Type that fraction in Desmos and tap its fraction button; it shows $-\frac{3}{7}$. So $a$, the squared-term coefficient of the filtration-efficiency model, is $-\frac{3}{7}$. Choice D.