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