Construct linear and exponential functions · lesson

Linear or exponential?

💡 Same difference each step means linear; same ratio each step means exponential.

Check the differences between outputs. If they're all the same, the pattern is linear.

If not, check the ratios (divide each output by the one before). If they're all the same, it is exponential.

🧠 Let's work it out

Is 3, 12, 48, 192 linear or exponential?

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💡 Find the slope from the two points, then the y-intercept, and write f(x) = mx + b.

Slope m = change in y ÷ change in x. Then use either point to solve for b.

A linear story is start + rate × time, which is the same form.

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Write the linear function through (2, 7) and (5, 16).

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💡 An exponential amount is start × factor^steps.

If an amount doubles each hour, multiply by 2 once per hour. After t hours it is start × 2^t.

Count the steps carefully. Day 0 is the start, so day 4 means 4 multiplications.

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200 cells double every day. How many after 4 days?

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💡 Growing by r percent multiplies by (1 + r) each step; shrinking by r percent multiplies by (1 − r).

Growing by 4% each year means keeping 100% and adding 4%, so multiply by 1.04 each year.

Losing 20% each year leaves 80%, so multiply by 0.80 each year.

Repeat once per year. Don't just add up the percents.

🧠 Let's work it out

$2,000 earns 4% interest compounded yearly. How much is there after 2 years?

Growing 4% means times 1.04 each year: 2000 × 1.04².
1.04² = 1.0816.
2000 × 1.0816 = 2163.20 dollars.
Decay works the same way: a 10,000-person town losing 20% a year has 10,000 × 0.8² = 6,400 after 2 years.

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💡 In f(x) = a · b^x, a is the value at x = 0 and b is the factor for each step of 1.

The point at x = 0 gives a directly, because b⁰ = 1.

To find b, divide the next output by the one before. If the points are 2 steps apart, b² is that ratio, so take the square root.

A factor between 0 and 1 means decay.

🧠 Let's work it out

Find f(x) = a · b^x through (0, 54) and (2, 6).

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