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Spot growth that multiplies each step

About 3 min

Adding the same amount and multiplying by the same factor can look alike at first. A third value can reveal the difference.

A garden model starts with 3 plants and doubles the count each season: 3, 6, 12, 24. A model that adds three starts 3, 6, 9, 12. The first two counts match. The third shows whether the rule adds or multiplies.

Turn doubling into a rule

Let x count seasons and y count plants. Find the count after two seasons.

  1. Record the start

    y=3×2x;x=0⇒y=3×1=3y=3\times2^x;\quad x=0\Rightarrow y=3\times1=3

    Here x counts seasons and y counts plants. Before any doubling, the count stays three. We write 2⁰ = 1 for zero steps.

  2. Read the power as repeated factors

    x=2⇒y=3×2×2=12x=2\Rightarrow y=3\times2\times2=12

    The exponent is the small raised number. It counts factors of two here. It does not mean two times the season number.

  3. Check one more season

    x=3⇒y=3×2×2×2=24x=3\Rightarrow y=3\times2\times2\times2=24

    Double twelve to get twenty-four. The added amount grows, but the multiplying factor stays at two.

The same factor makes growing jumps

The count starts at three. At one season it is six; at two it is twelve. The graph curves because the added amount grows.

Connect the pattern to the curve

A fixed factor at equal steps marks exponential growth. A fixed increase marks a linear rule. Check three values rather than just two if both patterns seem possible.

The graph places seasons along the bottom and plant counts up the side. As the added amount grows, the curve gets steeper. The whole-season counts give clear points to check.

The smooth curve also shows the rule between those points. It need not count whole plants there. Real plants cannot double forever: space and food run out. This is a model of a pattern.

Try it yourself

Work out an answer and explain your reasoning. Then check the explanation.

Use another growth factor

y=2⋅33y=2\cdot3^3

Find the count after three steps. Explain how the repeated change differs from adding three.

Show the answerHide the answer
2⋅3⋅3⋅3=542\cdot3\cdot3\cdot3=54

The values are 2, 6, 18 and 54.

Separate powers from straight lines

4⋅2x4+2x4\cdot2^x\qquad4+2x

Which rule doubles its result each step? Use enough values to tell the two patterns apart.

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4⋅2x4\cdot2^x

The first gives 4, 8, 16. The second gives 4, 6, 8 because it adds two each time.

Tell two matching starts apart

y=2×2xy=2\times2^x

Both give two at zero and four at one. What happens at the next step?

Graph A

The first three values are two, four and six.

Graph B

The first three values are two, four and eight.

Show the answerHide the answer
Graph B\text{Graph B}

Doubling gives eight at x = 2. Graph B fits. Graph A keeps adding two, so it gives six.

When an extra number shifts the curve

In y = 2ˣ + 3, add three after taking the power. At x = 0, 1 and 2, you get 4, 5 and 7. Those full results do not double. Calculate points from the full rule when it includes an extra shift.

In y = −2ˣ, the minus sign changes each height’s sign. The curve sits below zero. It does not turn the base negative.

Put the growth pattern to the test

Match rules to graphs. Check the start and a few later values. Include any extra multiplier or added number in the full rule.

2-minute practice · keep going after mistakes · opens in a new tab

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