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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 - PowerPoint PPT Presentation
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Chapter 5: Exponential and Chapter 5: Exponential and Logarithmic Logarithmic
FunctionsFunctions5.2: Exponential Functions5.2: Exponential Functions
Essential Question: In what ways can you translate an exponential function?
Graphs of Exponential Graphs of Exponential FunctionsFunctionsFor 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 the
negative side of the x-axis as x approaches -∞
f(x) = 2x
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
Graphs of Exponential Graphs of Exponential FunctionsFunctionsFor 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 the
positive side of the x-axis as x approaches ∞
f(x) = (½)x
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
Graphs of Exponential Graphs of Exponential FunctionsFunctions
Translations◦ Just like translations with other functions…
Changes next to the x (exponent) affect the graph horizontally, and opposite as expected
Examples: 2x+2 shifts the parent function 2x left 2 units 34x stretches the parent function 3x horizontally by a
factor of ¼ 2-x flips the parent function 2x horizontally
Changes away from the x affect the graph vertically, as expected
Examples: 4x + 5 shifts the parent function 4x up 5 units 7 • 2x stretches the parent function 2x vertically by a
factor of 7 -3x flips the parent function 3x vertically
Graphs of Exponential Graphs of Exponential FunctionsFunctionsExample
◦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 verticallyvertical stretch by a factor of 5horizontal shift right 1 unitvertical shift up 7 units
Graphs of Exponential Graphs of Exponential FunctionsFunctionsAssignment
◦Page 343 Problems 1-13 & 36-39 (all) Due tomorrow
◦What’s tomorrow? Word problems!!!
Graphs of Exponential Graphs of Exponential FunctionsFunctionsUsing 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
Graphs of Exponential Graphs of Exponential FunctionsFunctionsUsing 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
Graphs of Exponential Graphs of Exponential FunctionsFunctionsUsing 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
Graphs of Exponential Graphs of Exponential FunctionsFunctionsThe 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)
Graphs of Exponential Graphs of Exponential FunctionsFunctionsUsing 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
Graphs of Exponential Graphs of Exponential FunctionsFunctionsAssignment
◦Page 344 Problems 50-57 (all problems)
Ignore parts b and/or c from each problem
Due tomorrow