Question 44·SAT Desmos Regression Practice·Problem Solving and Data Analysis

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The table shows values of $x$ and $y$ for a set of measurements. The $y$ -values are rounded to the nearest whole number.

$x$	$y$
$1$	$510$
$4$	$313$
$7$	$193$
$10$	$118$

An exponential model of the form $y=a\,b^x$ is proposed, where $a>0$ and $0<b<1$ .

Which choice gives a model that is most consistent with the data?

Rounded measurements call for an exponential best fit, not an equation that hits every listed whole number. Put the paired values in a Desmos table and fit the stated form $y=ab^x$ , then compare its starting value and multiplier with the choices. The three-unit gaps between table rows can tempt you to use $x/3$ as the exponent, but that changes how often the multiplier acts.

Hints

Hint 1
An exponential regression finds a curve that follows all the measured pairs. Because the outputs were rounded, the best model need not reproduce every displayed whole number. What two numbers does the proposed form need?
Hint 2
Put each input in Desmos’s $x_1$ column and its paired output in $y_1$ . Fit $y_1\sim ab^{x_1}$ ; the tilde tells Desmos to estimate $a$ and $b$ using all four rows.
Hint 3
In $ab^x$ , $a$ is the initial value at $x=0$ , not the first output shown. The factor $b$ applies each time $x$ rises by one. How many such changes occur between table rows?

Step-by-step

Approach 1: Fit the stated model in Desmos
Step 1
Decide what kind of fit is needed
These are rounded measurements, so the requested model should follow the data closely, not necessarily produce every listed whole number exactly. That calls for a best fit using the exponential form the question gives.
Step 2
Fit the exponential form
Enter each table row as an $(x_1,y_1)$ pair. Then type $y_1\sim ab^{x_1}$ ; the tilde asks Desmos to fit the stated form. Under PARAMETERS, Desmos reports about $a=600.16$ and $b=0.8500$ . Look for a choice with a starting value near $600$ and a one-unit factor near $0.85$ .
Step 3
Check what the starting value means
The initial value $a$ belongs to $x=0$ , which the table doesn't show. Type $600(0.85)$ for $x=1$ ; Desmos shows $510$ , matching the first row. So $510$ is not the starting value: the factor has already been applied once.
Step 4
Compare predictions with the remaining rows
Type $600(0.85)^{[4,7,10]}$ to check the other inputs at once. Desmos shows about $[313.20,192.35,118.12]$ , close to the measured $[313,193,118]$ . The prediction at $x=7$ doesn't even round to $193$ ; most consistent doesn't mean an exact match to every measurement. The model is $y=600(0.85)^x$ . Choice A.

Approach 2: Find the one-unit factor from two rows
Step 1
Measure the change across one table gap
From $x=1$ to $x=4$ , the input rises by $3$ . Type $r=313/510$ ; Desmos shows $r\approx0.6137$ . This ratio estimates the factor for that three-unit gap, not for one unit.
Step 2
Convert the gap factor to a one-unit factor
Three $x$ -units mean three copies of the one-unit factor. So $r=b^3$ . Type $b=r^{1/3}$ ; Desmos shows $b\approx0.8498$ . The cube root finds the factor that, when multiplied by itself three times, gives $r$ .
Step 3
Work back to the starting value
At $x=1$ , the model gives $ab$ , which is about $510$ . Type $a=510/b$ ; Desmos shows $a\approx600.13$ . These two rows point to $a$ near $600$ and $b$ near $0.85$ , agreeing with the fit to all four rounded measurements. The matching model is $y=600(0.85)^x$ . Choice A.

Question 44·SAT Desmos Regression Practice·Problem Solving and Data Analysis

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Step-by-step

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Step-by-step

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