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B. Exponential Functions · Exponential Functions

Collecting Data That Can Be Modelled by an Exponential Function

B2.1 B2.2About 75 minutes
Learning goals
  • I can collect data from an experiment (such as removing coins or bouncing a ball) that can be modelled by an exponential function.
  • I can organize the data in a table, graph it, and use ratios to check if an exponential model fits.
  • I can build a model y = a(b)ˣ by hand, using the initial value and an average ratio (or two points and a rational exponent).
  • I can describe how technology (exponential regression) finds a curve of best fit, and where to find real data (secondary sources).

Words to know

primary data
Data you collect yourself, for example by doing an experiment and measuring.
secondary data
Data collected by someone else, for example tables from Statistics Canada or a science website.
model
An equation that describes data approximately and can be used to predict.
exponential regression
A technology tool that finds the exponential curve y = a(b)ˣ that fits a set of data points as closely as possible.
curve of best fit
A smooth curve that passes as close as possible to the data points.
discrete / continuous data
Discrete data can only take separate values (bounce number 1, 2, 3); continuous data can take any value in an interval (time, temperature).

Where do exponential patterns come from?

Exponential functions appear whenever something changes by the same fraction of itself in each time step:

  • each bounce of a ball rebounds to about the same fraction of the height before,
  • a hot drink loses about the same fraction of its extra heat each minute,
  • a population grows by about the same percent each year,
  • a radioactive sample loses about the same fraction of its atoms in each time interval.

In this lesson you will collect your own data, check that it is exponential, and build an equation for it. Real data are never perfect, so you will also learn how to judge "close enough".

Experiment 1: The coin-removal experiment

Put 50 coins in a cup, shake, and pour them onto a table. Remove every coin that lands heads. Count the coins left. Put them back in the cup and repeat.

Each coin has a 1/2 chance of being removed each shake, so on average half the coins remain each time. The theoretical model is N = 50(0.5)ᵗ, where t is the number of shakes. The data below are a sample result from one person's experiment; yours will be a little different, which is normal for chance experiments.

Shake t012345
Coins left N502614742
Ratio to previous0.520.540.500.570.50
Sample data. The ratios are all close to 0.5, so an exponential model fits well.
shakecoins left12345610203040500● sample data- - model N = 50(0.5)ᵗ
Sample coin data (points) and the model N = 50(0.5)ᵗ (dashed). The data are discrete (whole shakes only), so we plot points; the model curve is drawn to show the trend.

Building a model by hand

An exponential model y = a(b)ˣ needs two numbers:

  1. a, the initial value: the y-value when x = 0.
  2. b, the growth or decay factor: find it one of two ways.
  • Average ratio: find the ratio for each step and take their mean.
  • Two points: if y goes from a to yₙ in n steps, then a × bⁿ = yₙ, so b = (yₙ / a)^(1/n). This uses the rational exponents from earlier in the unit!

Then check the model: compare its predictions with the data. If they are close, the model is good.

With the coin data: the mean of the ratios is (0.52 + 0.54 + 0.50 + 0.57 + 0.50) ÷ 5 ≈ 0.53, giving N = 50(0.53)ᵗ. The two-point method with (0, 50) and (5, 2) gives b = (2/50)^(1/5) = 0.04^(1/5) ≈ 0.53 as well. Both are close to the theoretical 0.5.

Experiment 2: A bouncing ball

Drop a ball from 100 cm beside a wall with a measuring tape taped to it. Record the height of the top of each rebound. A phone video in slow motion makes the reading much easier. A sample result for one bouncy ball:

dropped from 100 cm72 cmbounce 152 cmbounce 237 cmbounce 327 cmbounce 419 cmbounce 5Each rebound is about 72% of the one before it (ratio ≈ 0.72).
Sample rebound heights for one ball (drawn roughly to scale). The heights shrink by the same ratio, not the same amount.
Worked example

Example 1: Model the bouncing ball

Sample data: bounce n = 0, 1, 2, 3, 4, 5 and height h = 100, 72, 52, 37, 27, 19 cm. (a) Show that an exponential model is reasonable. (b) Write a model. (c) Predict the height after bounce 7.

  1. (a) First differences −28, −20, −15, −10, −8 are not constant. Ratios: 72/100 = 0.72, 52/72 ≈ 0.722, 37/52 ≈ 0.712, 27/37 ≈ 0.730, 19/27 ≈ 0.704. All close to 0.72, so exponential is reasonable.
  2. (b) a = 100 (the drop height, n = 0) and b ≈ 0.72. Model: h = 100(0.72)ⁿ.
  3. Check: n = 3 gives 100(0.72)³ ≈ 37.3 (data 37 ✓); n = 5 gives ≈ 19.3 (data 19 ✓).
  4. (c) h = 100(0.72)⁷ ≈ 10.0 cm.

h = 100(0.72)ⁿ; about 10 cm after bounce 7.

Worked example

Example 2: The two-point method

A bacteria culture is counted at 0 h: 400 and at 6 h: 3200. Assuming exponential growth, find the hourly growth factor and write the model.

  1. a = 400. After n = 6 hours, 400 × b⁶ = 3200, so b⁶ = 8.
  2. b = 8^(1/6). Since 8 = 2³, 8^(1/6) = 2^(3/6) = 2^(1/2) = √2 ≈ 1.414.
  3. Model: N = 400(1.414)ᵗ, which means about 41.4% growth per hour. (Equivalently, the count doubles every 2 hours: N = 400(2)^(t/2).)

b ≈ 1.414; N = 400(1.414)ᵗ

Using technology: exponential regression

For more than a few points, technology finds the best-fitting exponential curve:

  • Desmos: enter the data in a table with columns x₁ and y₁, then type y₁ ~ a·b^x₁ in a new line. Desmos reports a, b and how well the curve fits.
  • Graphing calculator: enter the lists, then use the exponential regression command (ExpReg on many models).
  • Spreadsheet: make a scatter plot and add an exponential trendline. Some spreadsheets show the equation in an equivalent form that uses a special number called e; you will meet it in later courses.

Regression gives values very close to your hand-made model when the data are close to exponential. Always look at the graph to make sure the curve really follows the points.

Secondary sources

You do not have to collect every data set yourself. Statistics Canada publishes free data tables, for example population counts for provinces and cities over many years. Science websites and textbooks publish data such as the activity of radioactive samples over time. When you use secondary data, record the source (who published it, the table name, the date you accessed it), check that the time steps are equal before testing ratios, and remember that real populations only grow at a roughly steady percent for limited periods.

Just for fun

Funny but true: a coin has no memory. A coin that has survived five shakes is no more likely to stay than a brand-new one. Radioactive atoms behave the same way, which is why this coin experiment is a good model of radioactive decay. (Unlike atoms, though, your coins can go back in your pocket and be spent.)

Hands-on activity

Do one experiment yourself

Choose one experiment. Record your data in a table, then graph it on paper or in Desmos.

A. Coin removal. You need 30 to 50 coins and a cup. Shake, pour, remove the heads, count, repeat until none are left.

B. Bouncing ball. You need a bouncy ball, a tape measure taped to a wall, and a phone camera if possible. Drop from 100 cm and measure 4 or 5 rebound heights. Do this somewhere with a hard floor and nothing breakable nearby.

C. Die rolling. You need 20 or more dice (or one die rolled once per "atom" if you are patient). Roll them all and remove every 6. Repeat. The model is N = N₀(5/6)ᵗ.

For your data: (1) find the ratios; (2) write a model y = a(b)ˣ by hand; (3) predict one value you did not measure and test it if you can.

What to look for: coins: ratios near 0.5; ball: ratios that stay roughly the same (their value depends on the ball and the floor); dice: ratios near 5/6 ≈ 0.83. Small samples are noisy, so expect some ratios to be off. Ratios that are close to constant mean an exponential model is appropriate.

Your turn

1

Data: x = 0, 1, 2, 3; y = 200, 150, 112, 85. Find the ratios. Is an exponential model reasonable?

2

Write a model for the data in the last question.

3

In the coin experiment you start with 80 coins. Write the theoretical model.

4

In a die experiment you remove every die showing 5 or 6. What is the theoretical ratio?

5

A ball dropped from 2 m rebounds to 1.5 m. Assuming the same ratio each time, how high is the third rebound?

6

Is "number of bounces" discrete or continuous? How should you graph it?

7

Name one secondary source of exponential-looking data.

More practice: warm-up

1

What is a in y = a(b)ˣ?

2

What is b in y = a(b)ˣ?

3

Data: 10, 20, 40, 80. Ratio?

4

Data: 64, 32, 16, 8. Model if the first value is at x = 0?

5

Is temperature over time discrete or continuous?

More practice: on your own

1

Use the two-point method: y = 50 at x = 0 and y = 400 at x = 3. Find b and the model.

2

Use the two-point method: y = 1000 at x = 0 and y = 250 at x = 4. Find b to three decimal places.

3

Data: x = 0, 1, 2, 3, 4; y = 3, 5, 9, 16, 27. Find the ratios and the mean ratio (two decimal places). Write a model.

4

Check your model from the last question at x = 4. How close is it?

5

Why might a coin experiment with only 10 coins give ratios far from 0.5?

Challenge

1

Data (x = 0, 2, 4, 6): y = 5, 11.25, 25.3, 57.0. Find the growth factor per 1 unit of x and a model.

2

Two students model the same bouncing-ball data: h = 100(0.72)ⁿ and h = 100(0.7)ⁿ. Using the sample data (bounce 5: 19 cm), which fits better? Explain.

3

Design an experiment with a paper strip that produces exponential decay data with ratio 2/3. Describe what you measure.

Check your understanding

Quiz

1.

Which data set is best modelled by an exponential function? (equal x-steps)

2.

In y = a(b)ˣ fitted to data, how do you find a?

3.

A quantity goes from 20 to 160 in 3 equal steps, growing exponentially. What is b?

4.

In the coin-removal experiment (remove heads), what is the theoretical ratio?

5.

Which is a secondary source of data?

6.

A ball's rebound heights are 120, 90, 68, 51 cm. Which model fits best?

For parents and teachers

Big idea (B2.1, B2.2): students collect data that can be modelled by an exponential function, from their own experiments (primary data) or from sources such as Statistics Canada (secondary data), graph it, and recognize exponential patterns.

How to help: Do one of the experiments together; they take about 15 minutes. Recording a ball bounce in slow motion on a phone makes reading heights much easier. Talk about why the results are not perfect (chance, measurement).

Safety: use a clear area for bouncing balls, and only hot tap water (with an adult) if your teen tries a cooling experiment.

Common difficulties:

  • Expecting perfect data. Ratios that are roughly equal are enough.
  • Using differences instead of ratios to choose a model.
  • Forgetting a is the value at x = 0, not at x = 1.
  • The two-point method: b = (yₙ/a)^(1/n) needs the brackets on the calculator.