Question 11·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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The table represents points in a scatterplot.

$x$	$y$
$0$	$11$
$1$	$17$
$2$	$29$
$3$	$54$
$4$	$101$
$5$	$197$

A researcher believes that, except for one measurement error, the data follow an exponential model of the form $m(x)=a(b)^x+c$ , where $a$ , $b$ , and $c$ are positive constants. Which choice gives the $y$ -value that is most likely due to measurement error?

A shifted exponential model adds a fixed number to every output, so dividing consecutive outputs doesn't reveal its growth factor. With equally spaced inputs, subtract neighboring outputs and compare those changes instead. Use Desmos to list the changes, extend the pattern, and locate two adjacent breaks. One bad measurement at an interior input changes both differences that touch it.

Hints

Hint 1
In $m(x)=a(b)^x+c$ , the baseline $c$ is added to every output. It cancels when you subtract one output from the next. What changes do you get from the table?
Hint 2
An incorrect measurement at an interior input affects the difference before it and the difference after it. Look for two neighboring differences that break an otherwise steady multiplying pattern.
Hint 3
The first two differences give a growth factor. Extend that pattern and check it against the last difference. Which single table entry do the mismatched differences share?

Step-by-step

Compare consecutive changes
Step 1
List the changes in Desmos
Enter the six rows in a Desmos table. Then type $d=y_1[2...6]-y_1[1...5]$ . The brackets select entries $2$ through $6$ and $1$ through $5$ , so Desmos subtracts each output from the next and prints $d=[6,12,25,47,96]$ . These are the consecutive changes in $y$ .
Step 2
See why the changes should multiply
Subtract two outputs one input unit apart:
$m(x+1)-m(x)=(ab^{x+1}+c)-(ab^x+c).$
Cancel the added baseline $c$ :
$m(x+1)-m(x)=ab^{x+1}-ab^x.$
Factor out $ab^x$ :
$m(x+1)-m(x)=ab^x(b-1).$
Increasing $x$ by $1$ multiplies this change by $b$ . So compare the changes, not the original outputs; the added $c$ prevents the outputs themselves from having a constant ratio.
Step 3
Find the factor from the first two changes
The first two changes are $6$ and $12$ . Type $12/6$ in Desmos; it shows $2$ . So the growth factor suggested by those changes is $b=2$ .
Step 4
Locate the two broken changes
Type $6\cdot2^{[0...4]}$ to generate five changes by doubling. Desmos prints $[6,12,24,48,96]$ . The first two and the last change match the table, but $25$ and $47$ don't. The two mismatches are adjacent, so check the table entry they share rather than blaming either endpoint.
Step 5
Check the measurement at $x=3$
The two broken changes share the output at $x=3$ . The output at $x=4$ is $101$ , and the expected change from $x=3$ to $x=4$ is $48$ . Type $101-48$ ; Desmos shows $53$ . The table records $54$ instead of $53$ at $x=3$ , so $54$ is the $y$ -value most likely due to measurement error. Choice A.

Question 11·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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