Exponential Functions Section 3.1. What are Exponential Functions?

Exponential Functions

Section 3.1

What are Exponential Functions?

• Exponential functions are functions whose equations contain a variable in the exponent

• Categorized as Transcendental (Non-Algebraic)

• Examples:

f(x) = 2x g(x)=3x+1 h(x) = ( )x-1

Why study exponential functions?

• Many real-life situations can be described using exponential functions, including– Population Growth– Growth of epidemics– Radioactive decay– Compound Interest

Definition of Exponential Function

• The exponential function f with base a is defined by

f(x) = ax or y = ax

where a is a positive number other and 1 (a>0 and a ≠ 1) and x is any real number.

Exponential Functionf(x) = ax

• Domain: (-∞, ∞)• Range (0, ∞)• y-intercept: (0, 1)• NO zero (has a horizontal

asymptote at y=0)• Increasing (-∞, ∞)• No relative minimum or

maximum• Neither even nor odd• Continuous• Has an inverse (logarithm)

X y=f(x)-2-1012

f(x) = a(bx-c) + d

• Transformations learned in Chapter 1 still apply• Parent is exponential function with base a• Vertical translation –”d”• Horizontal translation –”bx-c=0” • Reflection on x-axis – “sign of a”• Reflection on y-axis-”sign of b”• Vertical Stretch or Shrink – “numeric value of a”

• EXAMPLES

Applications

• Compound Interest: A

• A = total (final) amount owed/earned• P=principal (initial amount

borrowed/deposited)• r= annual interest rate (%) must convert to decimal

• t= number of years • n= number of compoundings per year (“ly”)

Applications

• Example 1: You take out a loan of $30,000 to buy a new car. The bank loans you the

money at 7.5% annual interest for 5 years compounded monthly.

Applications• Example 2

You deposit $1 into an account paying 100% interest compounded: a) Yearlyb) semiannuallyc) quarterlyd) monthlye) weeklyf) dailyg) hourlyh) by the minutei) by the secondj) “continuously”

P r n A1 1 1 2annually

1 1 2 2.25semiannually

1 1 4 2.44140625quarterly

1 1 12 2.61303529monthly

1 1 52 2.69259695weekly

1 1 365 2.71456748daily

1 1 8760 2.71812669hourly

1 1 525600 2.71827924minute

1 1 31536000 2.71828178second

“e” ---Natural number

• An irrational number (lots of decimal places) • Denoted by “e” in honor of Leonard Euler• As n→∞, the approximate value of “e” to nine

decimal places is e ≈ 2.718281827…….

Applications

• Compound Interest for continuous compounding : A

• A = total (final) amount owed/earned• P=principal (initial amount

borrowed/deposited)• r= annual interest rate (%) must convert to decimal

• t= number of years

Applications

• Example 3: You invest $5000 for 10 years at an interest

rate of 6.5%. If continuous compounding occurs, how much money will you have in 10 years?

Applications

• Example 4 (Medicine):The radioactive substance iodine-131 is used in measuring heart, liver, and thyroid activity. The quantity, Q (in grams),

remaining t days after the element is purchased is given by the equation:

Calculate the amount remaining after 24 days.

Applications

• Example 5: A baby that weighs 6.375 pounds at birth may

increase her weight by 11% per month. Use the function:

where K is the initial birth weight, r is percent of increase (in decimal form), and t is time in months. How much would the baby weigh at 6 months?

Assignment

• Page 396 #25-31 odd, 35-45 odd, 53-55 odd, 65-67 odd, 73