4.1-Day 1

WARM-UP Your parents give you 3 options for an allowance plan when you are 1 year old. And you (the super protégé child) need to figure out which allowance plan to pick. First, write each of the three allowance options as a sequence mathematical function of your age, and tell me what type of model it is. Second, decide which allowance plan you’d pick. 1.You get 10 dollars a year when you are one and every year they add 10 dollars to your allowance, until age You get $1 dollar a year when you are one, when you are 2 you get $4, 3 you get $9, 4 you get $16, 5 you get $25, and so on until you are You get $2 dollar a year when you are one, then you get $4 the when you are 2, $8 when you are 3, $16 when you are 4….and so on until you are ten.

4.1—Exponential 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) = 2 x or f(x) = 3 -x.

Definition of an Exponential Function: The exponential function f with base a is denoted by: f(x) = a x where a > 0, a  1, and x (called the exponent) is any real number. What is the domain for any exponential function? Why is a  1 ? Why is a> 0? (Hint: What happens if a =-2?)

Rational Exponents 1. 2.

Negative Exponents 1. a -x =

Properties of Exponents

Ex. #1 Simplify the following without a calculator: a) 8 3 b) c) d)

Ex. #2 Make a table and graph the following without a calculator: a) f(x) = 2 x b) g(x) = 4 x x y x y

Ex. #3 Make an (x,y) chart and graph the following without a calculator: a) f(x) = 2 -x b) g(x) = x y x y

The five basic characteristics of typical exponential functions are listed below: a.) f(x) = b x b.) g(x) = b -x x y x y

Check Your answers The exponential function f with base a is denoted by: f(x) = a x, where a > 0, a  1, and x is any real number. The five basic characteristics of typical exponential functions are listed below: Graph of y = b x Graph of y = b -x*Domain:*Range: *y-intercept:1*y-intercept: 1 * Increasing function*Decreasing Function*Horizontal asymptote: y = 0

What is the relationship between the graph of the first and the graphs (i-iii).

f(x) = b  a x-c. b is vertical SHIFT c is the horizontal shift IF x is negative ---flip over x-axis

Ex. #4 Make an (x,y) chart and graph the following without a calculator: f(x) = 2 x a.) g(x) = 2 x+1 x y x y

Ex. #4 Make an (x,y) chart and graph the following without a calculator: f(x) = 2 x b) g(x) = 2 x - 3 x y x y

Ex. #4 Make an (x,y) chart and graph the following without a calculator: f(x) = 2 x b) g(x) = -2 x x y x y

4.1_Day 1 Summery The exponential function f with base a is denoted by: f(x) = a x, where a > 0, a  1, and x is any real number. The five basic characteristics of typical exponential functions are listed below: Graph of y = a x Graph of y = a -x*Domain:*Range: *y-intercept:1*y-intercept: 1 * Increasing function*Decreasing Function*Horizontal asymptote: y = 0 f(x) = b  a x-c. b is vertical or horizontal shift - means its reflected over the x-axis c is a phase shift

Graph one of the questions below. Q1: I’m still a little confused f(x) = -3 x-1 +2 Q2: I’m clear: An exponential function has a range of (2,- ∞ ), goes through the points (1,1) and (2, -1). Graph & write the equation! x y

Homework p ALL even & ALSO List out 5 basic characteristics of each graph!! Note: will need one sheet graph paper