A count that changes by the same percent each year shows exponential decay: each year multiplies the count by the same factor. Count the year-to-year steps, then use a Desmos regression to find that factor. The factor tells you what percent remains, not what percent was lost. Dividing the total loss by the number of years misses that each decrease acts on an updated count.
Hints
- Hint 1
A rate stated each year applies once from the start of one year to the start of the next. How many year-to-year changes happen between the two dates?
- Hint 2
A factor is the number you multiply a count by after each change. Applying the same factor twice multiplies the original count by its square. What equation matches the predicted count?
- Hint 3
The factor tells you what fraction remains after one year, not what fraction disappeared. Once you find it, subtract it from and multiply by to report the percent decrease.
Step-by-step
Solve for the yearly factor
Step 1Count the yearly changes
From the start of 2022 to the start of 2023 is one year. From there to the start of 2024 is another, so the predicted count changes by the same percent twice.
- Step 2
Write an equation for both years
Let be the yearly factor, the fraction of subscribers left after one year. Multiplying by it twice takes the starting count to the predicted count, so
Each year's percent acts on the updated count. The second decrease doesn't start from again.
- Step 3
Find the factor in Desmos
Type . The tells Desmos to find a value of that makes the sides match; the restriction keeps the factor between and because the count decreases but stays positive. Under PARAMETERS, Desmos shows . So of the subscribers remain each year.
- Step 4
Find the percent lost each year
The factor is what remains, so subtract it from to find what was lost. Type on the next Desmos line. It shows , meaning the predicted number of subscribers decreases by each year. Choice B.
Lessons that teach this
- SAT Nonlinear FunctionsIntermediateCoreBuild, identify, and interpret exponential models
- SAT Data AnalysisAdvancedReverse and combine percent changes
- SAT Data AnalysisIntermediateModel percent increase and decrease
- DesmosIntermediateCoreSolve percent problems in Desmos
- DesmosAdvancedCoreExponential models, transformations, and Log Mode