An exponential relation is one where, in a table of values, the ratio between consecutive outputs stays constant (rather than the difference staying constant, which is a sign of a linear relation). This constant ratio is the key giveaway: if each output is a fixed multiple of the one before it, you're looking at exponential growth (ratio greater than 1) or decay (ratio between 0 and 1). Choosing the right model for a real situation — linear, quadratic or exponential — starts with looking at how the values actually change: constant first differences suggest linear, constant second differences suggest quadratic, and a constant ratio suggests exponential.
Example
A population of bacteria doubling every hour produces values like 100, 200, 400, 800 — the ratio between consecutive values is always 2, confirming exponential growth, unlike a linear relation like 100, 200, 300, 400 where the difference (not ratio) stays constant at 100.
Key terms
Exponential relation:
A relationship where the ratio between consecutive output values stays constant.
Growth model:
A mathematical model (linear, quadratic or exponential) chosen to represent how a quantity changes.
Questions
1. In an exponential relation, the ratio between consecutive output values:
Stays constant
Is always zero
Always increases without limit
Is never a fixed number
2. A linear relation is identified by:
A constant difference between consecutive values
A constant ratio between consecutive values
No pattern at all
A constantly changing difference
3. Exponential growth occurs when the constant ratio is:
Greater than 1
Exactly 0
Exactly 1
Always negative
4. Exponential decay occurs when the constant ratio is:
Between 0 and 1
Greater than 1
Exactly 1
Always negative
5. The sequence 100, 200, 400, 800 has a constant ratio of:
2
100
0.5
400
6. A quadratic relation is identified by:
Constant second differences
A constant ratio between values
Constant first differences
No pattern whatsoever
7. Choosing between linear, quadratic and exponential models starts with:
Examining how the values actually change in a table
Guessing randomly with no evidence
Always assuming the relation is linear
Ignoring any data and picking a model arbitrarily
8. The sequence 3, 9, 27, 81 has a constant ratio of:
3
6
9
27
9. Is the sequence 5, 10, 20, 40 linear or exponential?
Exponential, with a constant ratio of 2
Linear, with a constant difference of 2
Neither linear nor exponential
Both linear and exponential simultaneously
10. Is the sequence 2, 5, 8, 11 linear or exponential?
Linear, with a constant difference of 3
Exponential, with a constant ratio of 3
Neither linear nor exponential
Both linear and exponential simultaneously
11. A car's value depreciating by 15% each year (rather than by a fixed dollar amount) is an example of:
Exponential decay
Linear decrease
Quadratic decrease
No mathematical pattern at all
12. A savings account earning a fixed dollar amount of interest each year (not a percentage) would best be modelled as:
Linear growth
Exponential growth
Quadratic growth
A relation with no identifiable pattern
13. Why might a phenomenon like compound interest be better modelled exponentially rather than linearly?
Growth is based on a constant percentage of the current amount, which produces a constant ratio, not a constant difference
Compound interest always produces a constant difference between values, matching a linear model
Exponential and linear models always describe compound interest identically well
The pattern of growth has no bearing on which mathematical model best fits it
14. Why might mistaking an exponentially growing quantity for a linear one lead to a significant underestimate of its future value?
Exponential growth accelerates over time, while linear growth increases by the same fixed amount each period, so a linear model would underestimate later values
Linear and exponential models always predict identical future values regardless of the underlying pattern
Exponential growth always predicts smaller future values than a linear model would
The choice of model has no bearing on how accurately future values are predicted
15. Why does a constant ratio (rather than a constant difference) between consecutive values indicate exponential rather than linear growth?
A constant ratio means each value is a fixed multiple of the previous one, which is the defining feature of exponential relationships
A constant ratio and a constant difference always indicate exactly the same type of relationship
Ratios between values have no mathematical connection to whether a relation is exponential
Constant ratios only ever appear in quadratic relationships, never exponential ones
16. Why might real-world population growth often be modelled exponentially in the short term, but not indefinitely into the future?
Exponential growth can approximate early, resource-unconstrained growth well, but real populations eventually face limits that slow growth
Population growth is always perfectly exponential with no limiting factors, indefinitely
Exponential models are never appropriate for modelling any real population, even in the short term
Resource constraints have no bearing on which mathematical model best fits population growth over time
17. Why might distinguishing between quadratic and exponential growth (both of which accelerate) require looking at second differences and ratios rather than just noting that both "curve upward"?
Quadratic growth has constant second differences while exponential growth has a constant ratio, so visual similarity alone isn't enough to tell them apart
Quadratic and exponential relations always produce completely identical graphs with no way to distinguish them
Any upward-curving graph is always exponential, never quadratic
Second differences and ratios provide no useful information for distinguishing between these two types of growth
18. Why might choosing the wrong growth model (like linear instead of exponential) for a real situation, such as disease spread, lead to seriously flawed predictions?
An exponentially growing quantity can increase far faster than a linear model would predict, especially over longer time periods
The choice of growth model never has any meaningful effect on the accuracy of real-world predictions
Linear and exponential models always produce identical predictions for any real-world situation
Disease spread is always more accurately modelled linearly than exponentially, without exception
19. Why might a table of values with only three or four data points sometimes make it genuinely difficult to confidently distinguish between a quadratic and an exponential pattern?
Both patterns curve upward and can look numerically similar over a small number of points, so more data may be needed to reliably tell them apart
Three or four data points is always more than enough to distinguish any type of pattern with complete certainty
Quadratic and exponential patterns always look completely different from each other, even with very few data points
The number of data points available has no bearing on how confidently a pattern can be identified
20. Why might a financial planner emphasise the importance of starting to save early, given how exponential (compound) growth behaves over long time periods?
Exponential growth accelerates over time, so money invested earlier has more compounding periods to benefit from that accelerating growth
The timing of when money is invested has no real bearing on how much it eventually grows under exponential compounding
Exponential growth behaves identically regardless of how much time is available for it to compound
Starting to save later always results in exactly the same outcome as starting earlier, under compound growth
21. Understanding exponential functions and growth models mainly helps you to:
Identify and choose the appropriate mathematical model based on how real values actually change
Assume every real-world relationship should always be modelled linearly
Ignore the difference between constant ratios and constant differences
Treat exponential and linear growth as producing identical long-term outcomes
Answer key (parent copy)
1. Stays constant
2. A constant difference between consecutive values
3. Greater than 1
4. Between 0 and 1
5. 2
6. Constant second differences
7. Examining how the values actually change in a table
8. 3
9. Exponential, with a constant ratio of 2
10. Linear, with a constant difference of 3
11. Exponential decay
12. Linear growth
13. Growth is based on a constant percentage of the current amount, which produces a constant ratio, not a constant difference
14. Exponential growth accelerates over time, while linear growth increases by the same fixed amount each period, so a linear model would underestimate later values
15. A constant ratio means each value is a fixed multiple of the previous one, which is the defining feature of exponential relationships
16. Exponential growth can approximate early, resource-unconstrained growth well, but real populations eventually face limits that slow growth
17. Quadratic growth has constant second differences while exponential growth has a constant ratio, so visual similarity alone isn't enough to tell them apart
18. An exponentially growing quantity can increase far faster than a linear model would predict, especially over longer time periods
19. Both patterns curve upward and can look numerically similar over a small number of points, so more data may be needed to reliably tell them apart
20. Exponential growth accelerates over time, so money invested earlier has more compounding periods to benefit from that accelerating growth
21. Identify and choose the appropriate mathematical model based on how real values actually change