The course has met six families: linear, quadratic, polynomial, rational, exponential and trigonometric. Modelling is choosing among them.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Testing from a table
Constant first differences mean linear. Constant second differences mean quadratic. Constant ratios mean exponential.
Repeating data
Values that rise and fall on a fixed schedule need a trigonometric model. No polynomial repeats forever.
Approaching a limit
Data levelling off toward a value suggests a rational or a decaying exponential, since those have horizontal asymptotes.
Context gives it away
Percentage change is exponential. A fixed amount per unit is linear. Area or projectile height is quadratic.
Check the fit
A model that matches the data but predicts nonsense outside it has been extrapolated too far.
Test the data for each family
Constant differences mean linear; constant ratios mean exponential; constant second differences mean quadratic. Checking a table in that order identifies the family or rules them all out.
Context suggests the family
Compound growth is exponential, projectile motion quadratic, steady rates linear, and repeating phenomena sinusoidal. Knowing what generates the data narrows the choice before any fitting.
Residuals judge the fit
A random scatter of residuals supports the model; a systematic pattern means the wrong family was chosen, however small the errors happen to be.
Every model has a range of validity
A model describes the data it was fitted to. Extrapolating well beyond that range assumes the relationship continues, which the data cannot support. Stating the range is part of the model.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Diego Identify a Family
A table gives 1, 4, 9, 16 for x = 1, 2, 3, 4.
- Step 1
The first differences are 3, 5 and 7, which are not constant.
Step 4: Your Turn
Practice makes it stick.
The Table
Problem 1 of 2
1, 4, 9, 16. Which family? 1 linear, 2 quadratic, 3 exponential.
The Growth
Problem 2 of 2
A balance growing 5% a year. Which family? 1 linear, 2 quadratic, 3 exponential.
Pick the Family
1 of 8
3, 7, 11, 15. Which family? 1 linear, 2 quadratic, 3 exponential.
2 of 8
2, 6, 18, 54. Which family? 1, 2 or 3?
3 of 8
1, 4, 9, 16. First differences are 3, 5, 7. What is the second difference?
4 of 8
Daily temperature over a year. Which family? 1 linear, 2 trigonometric.
5 of 8
A projectile height against time. 1 linear, 2 quadratic.
6 of 8
Total cost at a fixed price per item. 1 linear, 2 exponential.
7 of 8
Match each pattern with the family it identifies.
Tap a card on the left to start.
8 of 8
5, 10, 20, 40. Which family? 1 linear, 2 quadratic, 3 exponential.
Step 5: Quick Check
Show what you know.
Question 1 of 2
4, 9, 14, 19. Which family? 1 linear, 2 quadratic, 3 exponential.
Question 2 of 2
What identifies a quadratic from a table?
What You Learned
- Constant first differences mean linear, second differences quadratic, ratios exponential.
- Repeating data needs a trigonometric model.
- Context often names the family before any table is examined.