GRAPHING—Trigonometric Functions In algebra you studied “algebraic” functions such as polynomial and rational functions. In this chapter we will study two types of non-algebraic functions – exponential functions and logarithmic functions. These functions are called transcendental functions. Exponential functions are widely used in describing economic and physical phenomena such as compound interest, population growth, memory retention, and decay of radioactive material. Exponential functions have a constant base and a variable exponent such as f(x) = 2x or f(x) = 3-x. Definition of an Exponential Function: The exponential function f with base a is denoted by: f(x) = ax , where a > 0 , a  1, and x is any real number. What is the domain for any exponential function? ________________ Why is a  1 ? __________________________________________ Why must a > 0 ? RECALL: If you are not familiar with these please refer to pg 14 in your book under REVIEW OF BASIC ALGEBRA Rational Exponents 1. t 2. Negative Exponents a-x = Properties of Exponents 1.) 2. 3. Ex. #1 Simplify the following without a calculator: a) 83 b) c) d) Ex. #2 Make a table & graph the following without a calculator: a) f(x) = 2x b) g(x) = 4x x f(x) -2 -1 ½ 1 2 3 4 x -5 5 10 y 20 x -5 5 10 y 20 x y -2 -1 ½ 1 2 3 4 Ex. #3 Make an (x,y) chart and graph the following without a calculator: a) h(x) = 2-x b) t(x) = x -5 5 10 y 20 x -5 5 10 y 20 GRAPHING—Trigonometric Functions x f(x) -4 -3 -2 -1 -½ 1 2 Unit 5 x f(x) -4 -3 -2 -1 -½ 1 2 ** Note that comparing Ex.a) with Ex.b), we can write h(x) = 2-x = _______ and t(x) = = _________. The five basic characteristics of typical exponential functions are listed below: Graph of y = ax Graph of y = a-x *Domain: *Domain: *Range: *Range: *y-intercept: *y-intercept: *Inc. or dec. function? *Inc. or dec. function? *Horizontal asymptote: *Horizontal asymptote: Quickly sketch a graph- What is the relationship between the first graph and each other part (i-iii). We will use the graphs of ax to sketch graphs of functions in the form f(x) = b  ax+c. What does b do? What does c do? What if x is negative? Ex.#4 Sketch each of the following without a calculator: a) f(x) = 2x+1 b) h(x) = 2x - 3 c) g(x) = -2x *Want a second example: Read #4 in your book (section 4.1). In the definition on the first page we use an unspecified base a to introduce exponential functions. It happens that in many applications the convenient choice for a base is the irrational number e, where e = 2.71828…., called the NATURAL BASE. The function f(x) = ex is called the natural exponential function, where e is the constant and x is the variable. Ex.#5 a. Make an (x,y) chart and graph the following : f(x) = ex b. Where does e come from? Make an (x,y) chart and graph the following From this table, it seems reasonable to conclude that Pretend that you choose allowance option #1 yesterday, but you invested your first allowance check ($10) into a bank account. (Forget the other years payments for now). Next year your balance is $11, the following year it is $12.10, the following year is $13.31, then $14.64 (see table). What type of function is this? (linear? quadric? Exponential?) Do you see a pattern? Formulas for Compound Interest (see pg. 285) After t years, the balance A in an account with principal P and annual percentage rate r (expressed as a decimal) is given by the following formulas: 1. For n compoundings per year : yr 1 2 3 4 5 6 $ in bank $10 $11 $12.10 $13.31 $14.64 $16.11 $17.72 Check this 10.00 = 10(1.1)0 = 10(1+.1)0 11.00 = 10(1.1)1 = 10(1+.1)1 12.10 = 10(1.1)2 = 10(1+.1)2 13.31 = 10(1.1)3 = 10(1+.1)3 14.64 = 10(1.1)4 = 10(1+.1)4 16.11 = 10(1.1)5 = 10(1+.1)5 17.72 = 10(1.1)6 = 10(1+.1)6 P P(1+r)1 P(1+r)2 P(1+r)3 P(1+r)4 P(1+r)5 P(1+r)6 2. For continuous compounding : Ex.#6 A total of $12,000 is invested at an annual percentage rate of 9%. Find the balance after five years if it is compounded: a. quarterly b. Semiannually c. continuously (see ex 8 & 9 p.285) Ex.#7 Let y represent the mass of a particular radioactive element whose half-life is 25 years. After t years, the mass in grams is given by P P(1+r)1 P(1+r)2 P(1+r)3 P(1+r)4 P(1+r)5 P(1+r)6 Check this 10.00 = 10(1.1)0 = 10(1+.1)0 11.00 = 10(1.1)1 = 10(1+.1)1 12.10 = 10(1.1)2 = 10(1+.1)2 13.31 = 10(1.1)3 = 10(1+.1)3 14.64 = 10(1.1)4 = 10(1+.1)4 16.11 = 10(1.1)5 = 10(1+.1)5 17.72 = 10(1.1)6 = 10(1+.1)6 . a. What is the initial mass (when t = 0)? b. How much of the element is present after 80 years? (see ex 10 pg 286) Ex.#8 The number of fruit flies in an experimental population after t hours is given by Q(t) = 20e0.03 t ,t > 0. a. Find the initial number of fruit flies in the population b. How large is the population after 72 hours? c. Sketch a graph of Q(t) d. After how many hours would you expect to see about 300 fruit flies? See ex 11 pg 287

Each Slash mark is…. Period is… Amplitude

Each Slash mark is…. Period is… Amplitude

Homework Graphing Worksheet