What you will be able to do
- Linear patterns add a constant amount; exponential patterns multiply by a constant factor.
- In y = abˣ, a is the initial value at x = 0.
- b > 1 models growth; 0 < b < 1 models decay.
- Check equal input intervals before comparing output ratios.
Distinguish linear and exponential change
The linear outputs increase by 3. The exponential outputs multiply by 3. Constant difference and constant ratio are different structures.
| x | Linear y = 3x + 2 | Exponential y = 2·3ˣ |
|---|---|---|
| 0 | 2 | 2 |
| 1 | 5 | 6 |
| 2 | 8 | 18 |
| 3 | 11 | 54 |
Interpret y = abˣ
Worked example 1
A culture begins with 120 cells and doubles every hour: P(t) = 120·2ᵗ.
- 1.a = 120 is P(0), the initial value.
- 2.b = 2 is the growth factor per one-hour interval.
- 3.P(3) = 120·8.
Answer: P(3) = 960 cells
Check: 120, 240, 480, 960 follows repeated doubling.
Model exponential decay
Worked example 1
A value retains 80% each year and begins at $500.
- 1.The decay factor is 0.80, not 0.20.
- 2.Model V(t) = 500(0.80)ᵗ.
- 3.After 2 years: 500(0.64).
Answer: $320
Check: 500 → 400 → 320.
Read an exponential graph
- The y-intercept is the initial value a.
- Growth graphs rise increasingly quickly for positive a and b > 1.
- Decay graphs approach zero for positive a and 0 < b < 1.
- The graph remains continuous in many mathematical models, even if the context is sampled discretely.
- Units and domain restrictions determine which graph points are meaningful.
Practise representation changes
Move among a context, table, equation and graph. Ask what quantity is multiplied over an equal input interval and whether the initial value is visible.
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Useful answers
Frequently asked questions
How do I tell whether a table is exponential?
For equally spaced inputs, look for a constant nonzero ratio between consecutive outputs.
Is a percent decrease entered as the percent itself?
No. A 20% decrease retains 80%, so the factor is 0.80.
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Sources, scope and accuracy
The worked examples and practice questions on this page are original Cambridge Veritas learning materials, not released STAAR items. Curriculum and testing details can change; use the linked Texas Education Agency pages for current official information. Cambridge Veritas is not affiliated with or endorsed by TEA.