Grade 9 · Grade 9: Algebra, Quadratics and Functions · Algebraic Structure, Quadratics and Growth
Functions, Sequences and Exponential Growth
Use function notation, describe key graph features, and tell arithmetic, geometric and exponential growth apart.
Learning objectives
- Use function notation and interpret domain and range in context.
- Interpret intercepts, intervals of increase and decrease, maxima and minima.
- Write arithmetic and geometric sequences recursively and explicitly.
- Distinguish linear from exponential models and build each.
AERO Mathematics alignment
AERO.M9.AE.6
Use function notation, evaluate functions for inputs in their domains, and interpret statements about domain and range in context.
AERO.M9.AE.7
Interpret key features of graphs and tables, including intercepts, intervals of increase and decrease, maxima, minima and symmetry, for a function that models a relationship.
AERO.M9.AE.8
Write arithmetic and geometric sequences recursively and explicitly, and use them to model situations.
AERO.M9.AE.9
Distinguish between situations that can be modelled with linear functions and those modelled with exponential functions, and construct exponential models from tables, graphs or descriptions.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
Offer A pays 100 QAR on day 1 and 100 more each day. Offer B pays 1 QAR on day 1 and doubles each day.
Think about it
Which offer is better over 30 days, and when does the answer switch?
Hint: One grows by adding; the other grows by multiplying.
EXPLAIN
Functions, sequences and growth
Function notation
f(x) = 3x − 2 names the rule; f(4) = 10 is its output at 4. The domain is the set of allowed inputs and the range the set of possible outputs. In context, the domain is often restricted: negative time makes no sense.
Key features of a graph
- Intercepts: where the graph meets the axes.
- Intervals of increase and decrease.
- Maximum and minimum points.
- Symmetry and end behaviour.
Sequences
| Type | Recursive rule | Explicit rule | Example |
|---|---|---|---|
| Arithmetic | aₙ = aₙ₋₁ + d | aₙ = a₁ + (n − 1)d | 3, 7, 11, 15 with d = 4 |
| Geometric | aₙ = r · aₙ₋₁ | aₙ = a₁ · r^(n−1) | 3, 6, 12, 24 with r = 2 |
Linear or exponential?
Linear growth adds a constant each step; exponential growth multiplies by a constant each step. Offer A totals 46 500 QAR over 30 days, while Offer B pays over one billion on day 30 alone. Exponential growth always overtakes linear growth eventually.
Remember
Check equal steps in x. Constant differences mean linear; constant ratios mean exponential.
INVESTIGATION
Try it yourself
Model a savings plan two ways: adding 200 QAR a month, and increasing a 200 QAR balance by 8% a month. Tabulate 24 months, graph both, and find the month where the exponential plan overtakes the linear one.
Watch
Exponential Growth and Decay Word Problems
Khan Academy · 7:21
Contrasts adding a constant with multiplying by a constant.
Before you watch: Which grows faster in the long run: +5 each step or ×1.5 each step?
- Write an explicit rule for 3, 6, 12, 24, …
- Is a mobile plan with a fixed fee plus a per-minute rate linear or exponential?
Interactive simulation · PhET
Function Builder
Build a doubling machine and a plus-three machine and compare outputs after 10 steps.
While you explore
- Which machine grows faster in the long run, and why?
- How would you write each rule explicitly?
Key vocabulary
- Domain
- All the inputs a function is allowed to take.
- Geometric sequence
- A list where each term is multiplied by the same number.
Practice questions
0/1 correct
Level 1 · Criterion A
Which sequence is geometric?
MYP criterion tasks
Level 2 · Criterion B
A population of 500 grows 4% per year. Write the model and predict the population after 10 years.
Reflect & track
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