Year 10 Mathematics · AC9M10A03

Exponential Relations, Graphs and Equations

Exponential change is about multiplication, not simply “a graph that curves”. Learn to recognise constant ratios, read the meaning of y = abˣ, connect tables, rules and…

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Key conceptTeach from the board

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 examplesWe do

Worked examples

AC9M10A03 - Exponential Relations, Graphs and Equations
Example 1

Classify 8, 24, 72, 216 and state the ratio.

Example 2

Write a rule for an exponential relation with y=5 when x=0 and multiplier 4.

Example 3

Interpret y=600(0.7)ˣ in words.

Example 4

Place the solution of 2ˣ=7 between two consecutive integers before using a digital tool.

Curriculum examplesCopied content

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.

Lesson componentAustralian CurriculumVictoriaNSW
Recognise exponential patternsAC9M10A03VC2M10A11/A14 supportingMA5-NLI-C-01
Connect equation and graphAC9M10A03VC2M10A11MA5-NLI-C-01, C-02
Solve simple exponential equationsAC9M10A03VC2M10A14Non-linear content; MA5-LOG-P-01 only for advanced extension
Digital/approximate solvingAC9M10A03VC2M10A16 supportingMAO-WM-01 + relevant Path support
Questions and answersWith 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.
Practice and reviewReady for practice
  • “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.
Curriculum alignmentStart here
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