Introduction to Quantitative Methods Drawing exponential graphs

Paper Folding Activity Fold a piece of paper in half and record how many sections you have. Keep doing this until it gets too difficult to fold Record your results Number of folds Number of sections 1 2 3 4

Scattergraph Draw a scatter graph of the results. Start Where will the graph start? At the initial value 1

Formula If x is the number of folds and y is the number of sections can you come up with a formula connecting x and y? x y 1 2 4 3 8 × 2 × 2 × 2

Formula The multiplier is 2 1 = 20 (everything to the power 0 is 1) 2 = 21 4 = 22 8 = 23 So the number of sections = 2number of folds

Formula Putting in x and y The formula is y = 2x Initially there was 1 section This could be written as y = 1 × 2x

Drawing the graph using the formula In Excel it is easy to use a formula to make a table. The power is ^ Remember you can drag down once you have put in your first formula. A B C 1 =1*2^B1 2 =1*2^B2 3 =1*2^B3 4 =1*2^B4 5 =1*2^B5 6 =1*2^B6

Can you think of any other way to make the table in the spreadsheet rather than dragging the formula down?

Example In 1990 there were 285 mobile phone users in a area. The next year the number of mobile phone users had increased by 75%. How many mobile phone users would there be in 2000 if the number of users kept increasing by 75%? Who would be interested in knowing how many users there were in 2000?

Solution You could make a table, but because there are 10 years it would be quite big. Can you work out the formula if x is the number of years after 1990 and y is the number of mobile phone users?

Solution The Initial number of mobile phone users is 285. The multiplier is 1.75 The formula for year x would be y = 285 × (1.75)x In 2000 x = 10 so y = 285 × (1.75)10 = 76776

Excel tables Formulas Answers No of years after 1990 No of mobile phones 1 =285*(1.75)^A2 2 =285*(1.75)^A3 3 =285*(1.75)^A4 4 =285*(1.75)^A5 5 =285*(1.75)^A6 6 =285*(1.75)^A7 7 =285*(1.75)^A8 8 =285*(1.75)^A9 1 499 2 873 3 1527 4 2673 5 4678 6 8186 7 14326 8 25070

Graph Where will the graph start? At the Initial value 285

How do you think this compares with the actual number of mobile phone users? Do you think this is a realistic model?

Exponential Decay What if the numbers are decreasing….if we have exponential decay? What do you think the graph would look like? Try the following example and draw the graph on Excel.

Example The half-life of caffeine is about 6 hours. This means if you consume 200mg of caffeine at midday there will still be 100mg in your system at 6pm. This means that the amount of caffeine in your system halves every 6 hours. After 24 hours how much caffeine will be in your system?

Solution Let x be the number of half lives Let y be the amount of caffeine left in your system The initial value is 200 The multiplier is 0.5 The formula would be y = 200 × (0.5)x In 24 hours there are 4 half lives so x = 4 Amount left is 200 × (0.5)4 = 12.5mg

Graph Where will the graph start? At the initial value 200

Sketching graphs To sketch a graph all you need is to mark in the initial value and sketch the shape. Growth Decay Initial value Initial value

Summary Exponential growth or decay is of the form y = A(m)x Where A = Initial Value and m is the multiplier. You need to define x and y from your data If m > 1 it is growth if m < 1 it is decay

Spreadsheet On a spreadsheet the formula for exponential growth or decay is: And drag down. Where A = Initial Value and m is the multiplier A B C 1 =A*m^A1