1. Chapter 5: Exponential and. Logarithmic Functions 5
Chapter 5: Exponential and Logarithmic Functions 5.2: 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 the negative 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 the positive 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+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
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

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 →
-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 & (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 ≈ $

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) = x. Estimate the amount of plutonium remaining after 10,000 years.
Answer:
Let x = 10000
M(10000) = ≈ 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 = …
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) ≈ million people

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