Year 11 Standard · Unit 5 Ready for the Diploma · Lesson 1 of 6

What changes when a graph multiplies instead of adds?

A straight line adds the same amount for every step. An exponential function multiplies by the same factor for every step. This lesson asks what that one change does to a graph, and to a race between the two.

FunctionsScientific calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: in y = 3x + 2, find y when x = 4.
    Answer14. 3 × 4 = 12, and 12 + 2 = 14.
  2. From Unit 3 of this year: what is the common ratio of 5, 15, 45?
    Answer3. Each term is 3 times the one before.
  3. From Year 8: what is 2 × 2 × 2 × 2 × 2?
    Answer32. That is 2 to the power 5.
Words used in this lesson
exponential function
a function where y is multiplied by the same factor each time x goes up by 1. For example: y = a × bx.
base
the number that is raised to a power; here the factor for each step. For example: the 1.3 in 7 × 1.3x.
growth
values getting bigger because the base is above 1. For example: a population that multiplies each year.
decay
values getting smaller because the base is between 0 and 1. For example: how far a bowl of soup is above room temperature as it cools.
y-intercept
the value of y where the graph crosses the y-axis, which is a. For example: the start of a model.
exponent
the power, the count of multiplications. For example: the 2 in 52.

Lilies and reeds

A pond near Kochi has two things growing at its edge. A patch of lilies covers 1 square metre today and doubles its area every day. A bed of reeds also covers 1 square metre today, but it spreads by 12 square metres every day.

Predict first: the reeds will be far ahead after a day or two. Will the lilies ever catch them? If you think so, on roughly which day? Use the second figure to test your idea with smaller spreads, and the first figure to see what the numbers in the rule do.

2
1.2
0
2value of y at that x
growthkind of change

a = 2, b = 1.2, x = 0: y = 2. The base is above 1, so each step multiplies by 1.2 and the curve grows.

Show me the steps
  1. a = 2, b = 1.2, x = 0: the value is 2, the same as a, because 1.2 to the power 0 is 1. The base is above 1, so the curve grows.
  2. Slide x to 5: the value is 2 × 1.25, about 4.98. Every step multiplied the value by 1.2, so it rose by a different amount each step.
  3. Set a = 8, b = 0.5 and x = 3: the value is 8 × 0.53 = 1. The base is below 1, so each step halves the value: 8, 4, 2, 1. The curve gets close to 0 and never reaches it.

An exponential function multiplies by the same factor for every step.

  • It has the form y = a × bx, with a above 0 and b above 0 (and b not equal to 1).
  • a is the starting value: at x = 0, b0 = 1, so y = a. It is where the graph crosses the y-axis.
  • b is the factor for each step. If b is above 1 the values grow; if b is between 0 and 1 they decay.
  • The ratio of one y-value to the one before it is always b. For a straight line it is the difference that is always the same.

The curve never reaches the x-axis, and for x less than 0 it still has values: it is a function of every x.

1
2first whole day the lilies cover more
4 and 3lily area and reed area that day

Reeds spread 1 m² a day: on day 2 the lilies cover 4 m² and the reeds 3 m². The lilies are ahead from day 2.

Show me the steps
  1. Reeds spread 1 m² a day: on day 2 the lilies cover 4 m² and the reeds 3 m², so the lilies are ahead from day 2.
  2. Reeds spread 7 m² a day: the reeds lead for the first five days. On day 5 the lilies have 32 and the reeds 36, and on day 6 the lilies have 64 and the reeds 43.
  3. Reeds spread 9 m² a day: the reeds lead until day 5, and on day 6 the lilies cover 64 against 55. A larger spread delays the lilies; it does not stop them.

Power first, then multiply. 2 × 52 means 2 × 25 = 50. It does not mean (2 × 5)2 = 100. And a growing exponential can start behind a line: being ahead early says nothing about being ahead later. What decides it is the factor, not the starting position.

Worked example: bacteria in a dish

A culture starts with 800 cells and its number is multiplied by 1.5 every hour. Write the model and find the number of cells after 3 hours.

  1. The start is a = 800 and the factor is b = 1.5, so y = 800 × 1.5x, with x in hours.
  2. At x = 3 the power comes first: 1.53 = 1.5 × 1.5 × 1.5 = 3.375.
  3. Then multiply: 800 × 3.375 = 2700 cells.

The strength of a signal is y = 640 × 0.5x after x kilometres of cable. Is this growth or decay, and what is y when x = 4?

  1. The base 0.5 is between 0 and 1, so this is decay: the signal halves with each kilometre.
  2. 0.54 = 0.0625, and 640 × 0.0625 = 40.

Your go. The first step is given. The value of a machine is y = 250 × 0.8x thousand units after x years. Find its value after 3 years.

  1. The base is 0.8, so the value falls to 80% of itself each year.
  2. Work out 0.83 first, then multiply by 250.
Check the answer

0.83 = 0.512, and 250 × 0.512 = 128. A common slip is to multiply 250 × 0.8 × 3 = 600, which treats the exponent as a multiplier.

With a calculator

Work out the power first, then multiply. To find 6 × 1.43, type 1.4, press the power key (often marked xy), type 3 and press equals, then multiply by 6. Typing 6 × 1.4 first and then raising to the power 3 gives (6 × 1.4)3, which is a different number.

Check the size before you trust the display. 1.4 × 1.4 is about 2, so 1.43 is about 2 × 1.4 = 2.8, and the answer should be close to 6 × 2.8 = 16.8. The calculator gives 16.464.

Check before you go on

1. Two models are P: y = 5 × 3x and Q: y = 3 × 5x. Which is true at x = 0?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A model is y = 4 × 3x. Find y when x = 3.

3. A model is y = 96 × 0.5x. Find y when x = 5.

Level 2 · Building

4. A table gives y = 6, 9, 13.5, 20.25 for x = 0, 1, 2, 3, and the model is y = a × bx. Find b.

5. Which of these equations describes exponential decay?

Level 3 · Stretch

6. The model y = a × 4x passes through the point (2, 80). Find a.

7. Plan A starts at 10 and adds 6 each day. Plan B starts at 10 and doubles each day. After 6 days, how much more does Plan B have than Plan A?

Something to think about

The lilies in the pond double every day, and by day 30 they cover the whole pond. On which day did they cover half of it? What does that say about how quickly you notice an exponential change?

One way to think about it

Day 29. If the area doubles each day, the day before the pond is full it was half full. For most of the 30 days the patch looks small, and then it fills the pond in the last few days. An exponential change can seem slow right up until the end.

Exit ticket

8. A table gives y = 80, 40, 20, 10 for x = 0, 1, 2, 3. Which model fits?