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Comparing Exponential and Linear Functions

This lesson explores the differences between exponential and linear functions through a detailed comparison using a table of values. Students will learn to identify linear behavior through constant differences in values, while exponential growth is indicated by constant ratios. The lesson includes step-by-step instructions for entering data into a calculator, determining formulas for each function, and comparing growth rates. By solving equations graphically or algebraically, students will understand how exponential growth eventually outpaces linear growth.

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Comparing Exponential and Linear Functions

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  1. Comparing Exponential and Linear Functions Lesson 3.2

  2. Table of Values • Use the data matrix of your calculator to enter the following values. • Column 1 is the x values • Columns 2 and 3 are the f(x) and g(x)

  3. Table of Values • Click APPS then 6, then Current(F1, 8 to clear if needed) • Enter values as shown below

  4. Two Different Functions • Note the succession of values for f(x) • Differ by 6 each • This implies linear

  5. Cursor must be here Note ratio is 1.3 Two Different Functions • Note the succession of values for g(x) • They do NOT differ by a constant • However, the successive values have the same ratio • Show by placing formula in column 4 This shows it is an exponential function.

  6. How to Determine the Formula • From previous lesson we know that • Where b is the growth factor • In our case the common ratio = 1.3 • Solve for a • Substitute b = 1.3, any ordered pair

  7. Determine Function from Two Points • We can create twoequations with twounknowns • Solve for one of the variables • Substitute into the other equation • (-1, 2.5) (1, 1.6) • Answer b=0.8a=2

  8. Lineary = m*x + by = m+m+ … m+ b x terms Exponentialy = a * bxy = a * b * b * … * b x factors Compare and Contrast

  9. Compare and Contrast • Exponential growth will always outpace linear growth • In the long run • Eventually • Question:Where? • Solve graphically or algebraically or observe table

  10. Where Exponential Passes Linear • Giveny = 100 (1.02)x y = 100 + 12x • Algebraically : no current tools to solve 100 (1.02)x = 100 + 12x • Graphically : ask calculator to find intersection • Or observe tables

  11. Assignment • Lesson 3.2A • Page 119 • Exercises3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25 • Lesson 3.2B • Page 121 • Exercises27 – 39 odd

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