A dot plot shows every data value, so count positions to find the median, the middle of the ordered data. When every value is multiplied and then shifted, the median follows the signed rule, but the range, the distance from smallest to largest, uses the multiplier’s positive size. Read the middle and endpoints, then use a Desmos regression to solve the condition. A negative multiplier does not make the range negative.
Hints
- Hint 1
With 10 values, the median comes from the fifth and sixth values in order. Each dot counts as one value, even when several dots are stacked. Which number on the plot holds both middle positions?
- Hint 2
The range is largest minus smallest. Multiplying every value by a negative number reverses their order, but distances grow by the multiplier’s positive size. Adding the same to both ends does not change their distance.
- Hint 3
Apply the rule to the original median, then set the result equal to the new range. That gives an equation with one unknown, , for Desmos to fit.
Step-by-step
Transform the median and range
Step 1Find the middle of data set A
The median is the middle of the values in order. With 10 dots, use the fifth and sixth positions. The dot at and two dots at fill the first three positions; the three dots at fill positions four through six. So both middle values are , and A’s median is .
- Step 2
Find the range after the transformation
The plot’s smallest value is and its largest is . Multiplying by reverses their order but makes the gap three times as large; adding moves both ends equally. So B’s range, largest minus smallest, is . Type in Desmos; it shows .
- Step 3
Set the new median equal to the new range
Both middle values in A are . Transforming every value reverses their order, but those two values remain in the middle. So B’s median is . The median follows the signed rule; the range uses the positive stretch factor. Set the median equal to the range, as the question requires: .
- Step 4
Solve for the amount added
Type , replacing with so Desmos fits the unknown instead of making a slider. Under PARAMETERS, it shows . Adding to every transformed value makes B’s median equal its range. Choice C.
Lessons that teach this
- SAT Data AnalysisIntermediateCoreReason with range and standard deviation
- SAT Data AnalysisBeginnerFind and interpret center
- SAT Data AnalysisIntermediateRead distributions and data displays
- DesmosIntermediateCoreSpread, standard deviation, and box plots in Desmos
- DesmosIntermediateMean, median, and frequency tables in Desmos