Exponential Curve of Best Fit

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Key Ideas Easy Medium Hard Super Challenge

What to Remember

Constant ratio

If each y is multiplied by the same b when x goes up by 1, an exponential y = ab^x fits.

a is the start

When x = 0, y = a. If the table starts at x = 1, work backwards one step, or use y = ab^x with the first point.

Technology

An exponential regression on a calculator finds a and b that best fit scattered points. Check residuals and whether a constant ratio is reasonable.

Not every curve is exponential

A constant first difference is linear. A constant second difference is quadratic. Use exponential when ratios are nearly constant.

Level 1

Easy

A table with an exact constant ratio.

Example

The table is (0, 3), (1, 6), (2, 12), (3, 24). Write y = ab^x.

a = 3

Each output doubles, so b = 2.

y = 3 × 2^x

Answer: y = 3 × 2ˣ

The table is (0, 5), (1, 10), (2, 20). Write the exponential.

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y = 5 × 2^x

Answer: y = 5 × 2ˣ

The table is (0, 80), (1, 40), (2, 20). Write the exponential.

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b = 12

y = 8012^x

Answer: y = 80(1/2)ˣ

Does (0, 2), (1, 4), (2, 6) fit an exponential with integer b?

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Ratios 4/2 = 2 and 6/4 = 1.5, not constant. No. It is linear with slope 2.

Answer: no; linear

A culture is 10, then 30, then 90. Write N(t) = ab^t with t = 0 at the first count.

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a = 10, b = 3

N(t) = 10 × 3^t

Answer: N(t) = 10 × 3ᵗ

y = 7 × 2^x. What is y when x = 0 and when x = 3?

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y(0) = 7

y(3) = 7 × 8 = 56

Answer: 7 and 56

Level 2

Medium

The table may start at x = 1, or growth is a percent.

Example

A table is (1, 6), (2, 12), (3, 24). Write y = ab^x.

Ratio 2, so b = 2.

When x = 1, y = 6 = a × 2, so a = 3.

y = 3 × 2^x

Check x = 0 would be 3, one step before the table.

Answer: y = 3 × 2ˣ

A population is 400 and grows by 5% each year. Write P(t).

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P(t) = 400(1.05)^t

Answer: P(t) = 400(1.05)ᵗ

A car worth 20,000 dollars loses 10% of its value each year. Write V(t).

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V(t) = 20000(0.90)^t

Answer: V(t) = 20000(0.90)ᵗ

(0, 2), (1, 6), (2, 18), (3, 54). Write the model and find y(4).

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y = 2 × 3^x

y(4) = 2 × 81 = 162

Answer: y = 2 × 3ˣ; 162

Which model fits (0, 5), (1, 10), (2, 20) better: y = 5x + 5 or y = 5 × 2^x?

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Linear at x = 2 gives 15, but the point is 20.

Exponential gives 20. The exponential fits.

Answer: y = 5 × 2ˣ

A table of ratios is 2.1, 1.9, 2.0, 2.2. Would exponential regression be reasonable?

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The ratios cluster near 2, so yes, as an approximate fit. Not exact.

Answer: yes, approximately

Level 3

Hard

Compare models, or interpret a calculator fit.

Example

Points (0, 4), (1, 8), (2, 15), (3, 33). A linear fit and an exponential fit are both possible by eye. Which pattern do the ratios suggest, and what is a weakness of forcing y = ab^x?

Ratios: 8/4 = 2, 15/8 = 1.875, 33/15 = 2.2. Near 2, but not exact.

An exponential regression would smooth those ratios.

Weakness: the fit will miss at least one point, and with only four points a small miss can change a and b a lot.

Answer: near-exponential; not exact

A calculator reports y = 2.03(1.97)^x for a table that looks like doubling. Interpret 2.03 and 1.97.

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2.03 is a starting value near 2.

1.97 is a growth factor near 2 (almost doubling).

Rounding and scatter pulled the exact 2 × 2^x off a little.

Answer: start ≈ 2; factor ≈ 2

Data: (0, 100), (1, 50), (2, 25), (3, 12). Write an exact model if the last point were 12.5, then say what 12 instead of 12.5 means.

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Exact half-life: y = 100(0.5)^x

y(3) = 12.5

12 is a little below. Measurement noise, or the ratio is not exactly 1/2.

Answer: y = 100(0.5)ˣ if exact

Why is y = 3x + 1 a poor fit for 3, 9, 27, 81?

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First differences 6, 18, 54 are not constant.

Ratios are 3. Exponential y = 3 × 3^x with a shift of index, or y = 3^x if the first term is x = 1.

Answer: ratios constant, not differences

A scatterplot of years vs population looks curved up, with residuals from a line all positive in the middle. Why try exponential?

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A line underfits the middle and the end if growth is speeding up.

Exponential growth has an increasing slope. Regression can test whether a nearly constant ratio fits better.

Answer: growth is accelerating

Write two exponential models through (0, 5): one with b = 2 and one with b = 3. How do you decide with more data?

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y = 5 × 2^x and y = 5 × 3^x

The next point decides. If (1, 10), b = 2. If (1, 15), b = 3.

Answer: need another point

Super Challenge

A table is (0, a), (1, ar), (2, ar²). Show that the ratio of consecutive y-values is r, and write the function.

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ara = r, ar²ar = r

y = ar^x

Answer: y = arˣ

You run exponential regression and get b = 0.97. Is the quantity growing or decaying, and about what percent each step?

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b < 1, so decaying.

1 − 0.97 = 0.03, about 3% decay per step.

Answer: decay ≈ 3%

Give one reason an exponential curve of best fit can still be a bad model even if it looks close on the window you graphed.

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It may fail outside the data (extrapolation).

Or the real process cannot stay exponential (limited space, a linear phase, a drop to zero).

Or a quadratic or piecewise model matches residuals better.

Answer: extrapolation or a better family

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