1. LSP 120: Quantitative Reasoning and Technological Literacy Topic 2: Exponential Models Lecture notes 2.1 Prepared by Ozlem Elgun1

2. Exponential Change Like linear relationships, exponential relationships describe a specific pattern of change between a dependent and independent variable Recognizing and modeling this pattern allows us to make predictions about future values based on an exponential rate of change. While in linear relationships, y changes by a fixed absolute amount for each change in x (such as adding 3 each time), In exponential relationships, y changes by a fixed relative amount for each change in x (such as adding 3% each time). An exponential relationship is one in which for a fixed change in x, there is a fixed percent change in y. Prepared by Ozlem Elgun2

3. HOW TO CALCULATE PERCENT CHANGE We can use Excel to determine if a relationship is exponential by filling the neighboring column with the percent change from one Y to the next. As with linear, do not put this formula in the cell next to the first pair of numbers but in the cell next to the second. Prepared by Ozlem Elgun3

4. How to Format Cells in Excel Right click on the cell you want to format Click on ‘Format Cells…’ Select the ‘Number’ Tab Choose the appropriate format you want – i.e. If the cell contains a number, select ‘Number’ Display at least two decimal points Prepared by Ozlem Elgun4

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6. You can display percent change in decimal or percent format. For descriptive or visual representations you want to use the percent format. For calculations you want to use the decimal format. Above, I explain how to change the format of your cells. An easy way to format your cells to display percentages is to click on the ‘%’ icon on Excel. If the column is constant, then the relationship is exponential. So this function is exponential. Prepared by Ozlem Elgun6

7. Which of the following are exponential? Prepared by Ozlem Elgun7

8. Exponential function equation As with linear, there is a general equation for exponential functions. The equation for an exponential relationship is: y = P*(1+r) x P = initial value (value of y when x = 0), r is the percent change (written as a decimal), x is the input variable (usually units of time), Y is the output variable (i.e. population) The equation for the above example would be y = 192 * (1- 0.5) x Or y = 192 * 0.5 x. We can use this equation to find values for y if given an x value. Write the exponential equation for other examples. Prepared by Ozlem Elgun8

9. Why are exponential relationships important? Where do we encounter them? Populations tend to growth exponentially not linearly When an object cools (e.g., a pot of soup on the dinner table), the temperature decreases exponentially toward the ambient temperature (the surrounding temperature) Radioactive substances decay exponentially Bacteria populations grow exponentially Money in a savings account with at a fixed rate of interest increases exponentially Viruses and even rumors tend to spread exponentially through a population (at first) Anything that doubles, triples, halves over a certain amount of time Anything that increases or decreases by a percent Prepared by Ozlem Elgun9

10. If a quantity changes by a fixed percentage, it grows or decays exponentially. Prepared by Ozlem Elgun10