Exponential Relations, Graphs and Equations — AC9M10A03

What you will learn

• recognise exponential relations from constant ratios
• distinguish exponential change from linear and quadratic change
• interpret the parameters in y = abˣ
• connect a rule, a table and the corresponding graph
• solve simple exponential equations by using a common base
• use a table, graph or digital tool for approximate solutions when an exact method is not available

Prerequisite knowledge

You should know exponent laws, substitution, coordinates and basic graph interpretation. You should also recognise linear relations from a constant first difference. This lesson does not require logarithms; logarithms are a later/Path method for solving more difficult exponential equations.

Concept teaching

1. The constant-ratio test

When x increases by equal steps, an exponential relation multiplies y by the same factor. The values 5, 10, 20, 40 have constant ratio 2. The values 5, 10, 15, 20 have constant difference 5, so they are linear instead.

2. Read y = abˣ

For y = abˣ, a is the value when x = 0 because b⁰ = 1. The number b is the multiplier for each increase of 1 in x. If b > 1, values grow. If 0 < b < 1, values decay.

3. Percentage change and multipliers

A 7% increase means multiply by 1.07, not 0.07. A 7% decrease means retain 93%, so multiply by 0.93. Repeated percentage change is exponential because the percentage is applied to a changing amount.

4. Graph features carry algebraic meaning

For a positive a and b, the graph passes through (0,a). In a basic growth model y = abˣ with a>0 and b>1, the curve rises increasingly quickly; for 0<b<1, it falls towards 0. The algebraic multiplier and graphical shape are two representations of the same relationship.

5. Solving exponential equations

If both sides can be written with the same base, equate exponents. For equations such as 2ˣ = 10, a Year 10 method can use a graph, table or digital solver and report an approximate solution. Do not introduce logarithms as though they are required by this descriptor.

Worked examples

Common misconceptions and corrections

• “Any curved graph is exponential.” Quadratics and other functions also curve. Check the pattern or rule.
• “A 20% increase uses multiplier 0.20.” It uses 1.20; 0.20 is only the added portion.
• “a is the growth factor in y=abˣ.” a is the initial value; b is the per-step multiplier.
• “Constant difference means exponential.” Constant difference signals linear change; constant ratio signals exponential change.
• “Every exponential equation should be solved with logarithms.” At this level, common-base reasoning and digital methods are often the intended tools.
• “A calculator decimal is exact.” Digital solutions to equations such as 2ˣ=10 are approximate unless an exact representation is available.

Guided practice

1. Classify 8, 24, 72, 216 and state the ratio.
2. Write a rule for an exponential relation with y=5 when x=0 and multiplier 4.
3. Interpret y=600(0.7)ˣ in words.
4. Solve 5·3ˣ=135.
5. Place the solution of 2ˣ=7 between two consecutive integers before using a digital tool.

Independent practice

1. Decide whether 4, 12, 36, 108 is linear, quadratic or exponential.
2. Decide whether 3, 9, 19, 33 is exponential; justify your answer using differences or ratios.
3. Write y=abˣ for initial value 250 and 6% growth.
4. Write a decay rule for initial value 900 and 15% decrease per period.
5. Solve 4ˣ=64.
6. Solve 7·2ˣ=112.
7. Use a digital method to approximate the solution of 3ˣ=20 to 3 d.p.
8. For y=80(1.25)ˣ, find y when x=4 and interpret the result.
9. State two graph features of y=5(0.6)ˣ.
10. Explain why constant-percentage change creates a constant ratio.

Reasoning and problem-solving task

Two online videos start with 1000 views. Video A gains 600 views per day. Video B grows by 35% per day.

1. Write a linear model for A and an exponential model for B.
2. Calculate both models for days 0 to 5.
3. Identify when B first exceeds A.
4. Explain why the two models can initially look similar but diverge later.
5. Give two reasons neither model should be trusted indefinitely.

Important questions and answers

How can I recognise an exponential table?

For equal x-steps, the ratio of consecutive y-values is constant.

What does b mean in y=abˣ?

It is the multiplicative change for each one-unit increase in x.

What does a mean?

It is the value at x=0.

How do I solve a simple exponential equation?

First isolate the exponential term. If both sides can use the same base, equate exponents; otherwise use an appropriate graph/table/digital method.

Why use graphs?

They make growth/decay behaviour visible and allow an equation f(x)=g(x) to be interpreted as an intersection.

Assessment-style questions

1. 3 marks: A table has y-values 12, 18, 27, 40.5 for equal x-steps. Show that the relation is exponential and write its multiplier.
2. 4 marks: For y=320(0.84)ˣ, state the initial value, percentage change and whether the graph represents growth or decay.
3. 4 marks: Solve 5·2ˣ=160 exactly.
4. 5 marks: Use a graph/table/digital method to solve 1.08ˣ=2 approximately. State why your answer is approximate.
5. 6 marks: Compare L=500+120t with E=500(1.12)ᵗ over a suitable range of t. Explain one mathematical and one contextual difference between the models.

Review hint: When asked to “connect representations”, do not just calculate. Explain how a feature in the table or equation appears on the graph.

Exit ticket: mastery check

1. What table feature signals an exponential relation?
2. What percentage growth does multiplier 1.18 represent?
3. Solve 3·4ˣ=192.
4. Explain what the point (0,7) means on y=7·2ˣ.
5. Why might a graphing tool be appropriate for 2ˣ=11?

Teacher and parent guidance

For teachers

Move repeatedly among table, equation, graph and context. The common weakness is procedural recognition without representational connection. Give near-miss tables where differences are constant but ratios are not, and require students to justify classification. Treat logarithmic solving as optional extension aligned to local pathways, not as prerequisite content for this page.

For parents and carers

Ask “Is the amount being added each time, or multiplied each time?” That question often separates linear from exponential thinking. When a percentage changes repeatedly, encourage the student to convert it into a multiplier before calculating.

Curriculum alignment

Australian Curriculum: AC9M10A03, Year 10 Algebra — connecting algebraic and graphical representations of exponential relations and solving related exponential equations, using digital tools where appropriate.

Victoria: VC2M10A11, Level 10 Algebra supports connections between algebraic and graphical representations including exponential relations; VC2M10A14 covers simple exponential equations. VC2M10A16 can support numerical/graphical solving with digital tools.

NSW: MA5-NLI-C-01 identifies connections between algebraic and graphical representations of quadratic and exponential relationships; MA5-NLI-C-02 compares features of parabolas and exponential curves. MA5-LOG-P-01 is a supporting advanced Path outcome for logarithmic techniques, but logarithms are not presented here as required Australian Curriculum content.

Alignment explanation: Constant-ratio recognition, parameter interpretation and graph/equation connections directly support the Australian, Victorian and NSW non-linear relationship expectations. Simple equation-solving aligns strongly with Victoria A14; NSW may extend equation solving through advanced Path content, so that extension is clearly separated from the core lesson.

Practice and teaching resources

Related Year 10 Mathematics topics

• AC9M10A01 — algebraic expressions and equations
• AC9M10A04 — growth and decay modelling
• AC9M10N01 — exact values and approximation
• All Year 10 Mathematics topics

Official curriculum references

• ACARA — Australian Curriculum v9 Mathematics
• VCAA — Mathematics Version 2.0, Level 10
• NSW Curriculum/NESA — Mathematics K–10 (2022), Stage 5