For equally spaced inputs, check the outputs. Constant differences mean linear; constant ratios mean exponential.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Differences
The sequence 3, 7, 11, 15 adds 4 each time. Constant difference means linear.
Ratios
The sequence 3, 6, 12, 24 doubles each time. Constant ratio means exponential.
The early stage misleads
Over a short range an exponential can look almost straight. Checking ratios rather than eyeballing the graph avoids the trap.
The long run
Exponential growth eventually exceeds any linear growth, however steep the line.
Context helps
Percentage change of any kind is exponential. A fixed amount per unit is linear.
Differences against ratios
Linear data have constant differences between successive outputs; exponential data have constant ratios. Checking both in a table identifies the model, and often shows that neither fits.
Shape on a graph
Linear is straight; exponential curves increasingly steeply or flattens towards an asymptote. Over a short interval the two can look similar, which is why the table test is more reliable.
Choosing wrongly is costly
Treating exponential growth as linear badly underestimates the future. That error is behind persistent surprise at compound interest and at the speed of epidemic growth.
Some data are neither
Growth that slows towards a ceiling is neither linear nor exponential. Recognising that a model does not fit is a legitimate and valuable conclusion rather than a failure to choose.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Ines Classify a Table
A table gives 5, 10, 20, 40 for x = 0, 1, 2, 3.
- Step 1
The differences are 5, 10 and 20, which are not constant.
Step 4: Your Turn
Practice makes it stick.
The Table
Problem 1 of 2
5, 10, 20, 40. Is it linear or exponential? 1 linear, 2 exponential.
The Other Table
Problem 2 of 2
4, 9, 14, 19. Linear or exponential? 1 or 2.
Which Model
1 of 8
2, 6, 18, 54. Linear or exponential? 1 or 2.
2 of 8
7, 12, 17, 22. Linear or exponential? 1 or 2.
3 of 8
2, 6, 18, 54. What is the common ratio?
4 of 8
7, 12, 17, 22. What is the common difference?
5 of 8
100, 50, 25, 12.5. What is the common ratio?
6 of 8
A savings account paying 4% a year. Linear or exponential? 1 or 2.
7 of 8
Which signal identifies an exponential relationship?
8 of 8
3, 9, 27, 81. What is the common ratio?
Step 5: Quick Check
Show what you know.
Question 1 of 2
6, 12, 24, 48. Linear or exponential? 1 or 2.
Question 2 of 2
What check distinguishes the two models in a table?
What You Learned
- Constant differences mean linear; constant ratios mean exponential.
- An exponential can look straight over a short range, so check the ratios.
- Any percentage change is exponential.