Question 13·SAT Desmos Regression Practice·Advanced Math

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An app’s daily download count $D(t)$ (in thousands) is modeled by a quadratic function of $t$ , where $t$ is the number of days after the app is featured.

The model satisfies the following data:

$D(0)=1$
$D(3)=13$
$D(6)=7$

According to the model, which choice is the maximum value of $D(t)$ ?

Three input-output values for a quadratic model determine one parabola, even when its equation isn't given. Put the pairs in a Desmos table and run quadratic regression, then graph the fitted function and click its vertex, or turning point. If the parabola opens down, the vertex's output is the maximum, which may be higher than every listed output.

Hints

Hint 1
A function value pairs an input with its output: $D(3)=13$ gives the point $(3,13)$ . Write each condition as a point, keeping the day first.
Hint 2
A quadratic regression fits $ax^2+bx+c$ to the points you enter in a Desmos table. It reports the three coefficients together, so you don't have to solve three equations by hand.
Hint 3
The vertex is a parabola's turning point. If the coefficient of the squared term is negative, the curve opens down. At its peak, which coordinate gives the download count?

Step-by-step

Fit the model and click its peak
Step 1
Turn the function values into points
Each $D(t)$ pairs a day, the input, with a download count in thousands, the output. So the given values become $(0,1)$ , $(3,13)$ , and $(6,7)$ . Keep the day first in each pair.
Step 2
Enter the three pairs in a table
Enter the days in Desmos's $x_1$ column and their matching counts in its $y_1$ column. Check that Desmos shows $3$ and $13$ in the same row; switching the columns would model days from downloads instead.
Step 3
Fit the quadratic
Type $y_1\sim ax_1^2+bx_1+c$ . The tilde tells Desmos to fit a quadratic to the three rows. Under PARAMETERS, Desmos shows $a=-1$ , $b=7$ , and $c=1$ , giving $D(t)=-t^2+7t+1$ . Since $a$ is negative, the curve opens down and has a maximum at its peak.
Step 4
Read the maximum from the graph
Type $D(x)=ax^2+bx+c$ , using the fitted letters and letting $x$ stand for days. Click the peak of the graph; Desmos labels the vertex $(3.5,13.25)$ . The second coordinate is the maximum output, not the day it occurs: $13.25=\frac{53}{4}$ thousand downloads. For a downward-opening quadratic, the vertex's output is the maximum. Choice C.

Question 13·SAT Desmos Regression Practice·Advanced Math

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