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Chapter 5: Exponential and Logarithmic Functions 5.2: Exponential Functions

Chapter 5: Exponential and Logarithmic Functions 5.2: Exponential Functions. Essential Question: In what ways can you translate an exponential function?. Graphs of Exponential Functions. For an exponential function f(x) = a x If a > 1 graph is above x-axis y-intercept is 1

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Chapter 5: Exponential and Logarithmic Functions 5.2: Exponential Functions

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  1. Chapter 5: Exponential and Logarithmic Functions5.2: Exponential Functions Essential Question: In what ways can you translate an exponential function?

  2. Graphs of Exponential Functions • For an exponential function f(x) = ax • If a > 1 • graph is above x-axis • y-intercept is 1 • f(x) is increasing • f(x) approaches thenegative side of the x-axis as x approaches -∞ f(x) = 2x

  3. Graphs of Exponential Functions • For an exponential function f(x) = ax • If 0 < a < 1 • graph is above x-axis • y-intercept is 1 • f(x) is decreasing • f(x) approaches thepositive side of the x-axis as x approaches ∞ f(x) = (½)x

  4. Graphs of Exponential Functions • Translations • Just like translations with other functions… • Changes next to the x (exponent) affect the graph horizontally, and opposite as expected • Examples: • 2x+2shifts the parent function 2xleft 2 units • 34x stretches the parent function 3xhorizontally by a factor of ¼ • 2-x flips the parent function 2xhorizontally • Changes away from the x affect the graph vertically, as expected • Examples: • 4x+ 5 shifts the parent function 4xup 5 units • 7 • 2x stretches the parent function 2xvertically by a factor of 7 • -3xflips the parent function 3xvertically

  5. Graphs of Exponential Functions • Example • Describe the transformations needed to translate the graph of h(x) = 2x into the graph of the given function. • g(x) = -5(2x-1) + 7 • I’m not going to make you give me these in any order… anything not part of the parent function will change the graph • -5→ • -5→ • -1 → • +7→ flips graph vertically vertical stretch by a factor of 5 horizontal shift right 1 unit vertical shift up 7 units

  6. Graphs of Exponential Functions • Assignment • Page 343 • Problems 1-13 & 36-39 (all) • Due tomorrow • What’s tomorrow? • Word problems!!!

  7. Graphs of Exponential Functions • Using exponential functions • Example 4: Finances • If you invest $5000 in a stock that is increasing in value at the rate of 3% per year, then the value of your stock is given by the function f(x) = 5000(1.03)x, where x is measured in years. • Assuming that the value of your stock continues growing at this rate, how much will your investment be worth in 4 years? • Answer: • Let x = 4 • f(4) = 5000(1.03)4 ≈ $5627.54

  8. Graphs of Exponential Functions • Using exponential functions • Example 5: Population Growth • Based on data from the past 50 years, the world population, in billions, can be approximated by the function g(x) = 2.5(1.0185)x, where x = 0 corresponds to 1950. • Estimate the world population in 2015. • Answer: • Let x = 2015 – 1950 = 65, • g(65) = 2.5(1.0185)65 ≈ 8.23 billion

  9. Graphs of Exponential Functions • Using exponential functions • Example 6: Radioactive Decay • The amount from one kilogram of plutonium (239Pu) that remains after x years can be approximated by the function M(x) = 0.99997x. Estimate the amount of plutonium remaining after 10,000 years. • Answer: • Let x = 10000 • M(10000) = 0.9999710000 ≈ 0.74 kg

  10. Graphs of Exponential Functions • The Number e and the Natural Exponential Function • e is an irrational number, like π, which arises naturally in a variety of ways and plays a role in mathematical descriptions of the physical universe. You’ll explore the features of ex further in calculus. • e = 2.718281828459045… • ex is found on your calculator by pressing the 2nd button, followed by the ln key (one above x2)

  11. Graphs of Exponential Functions • Using exponential functions • Example 7: Population Growth • If the population of the United States continues to grow as it has since 1980, then the approximate population, in millions, of the United States in year t, where t = 0 corresponds to the year 1980, will be given by the function P(t) = 227e0.0093t. • Estimate the population in 2015 • Answer: • Let x = 2015 – 1980 = 35. • P(35) = 227e0.0093(35) ≈ 314.3 million people

  12. Graphs of Exponential Functions • Assignment • Page 344 • Problems 50-57 (all problems) • Ignore parts b and/or c from each problem • Due tomorrow

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