Functions & Graphs (1.2) What is a function? Review Domain & Range Boundedness Open & Closed Intervals Distance from a point to a line

Even & Odd Functions... Ex: Identify the domain, range, (use interval notation) and whether the function is odd or even or neither. y = x 2 y = √(1-x 2 ) y = √x y = 1/x y = 2x/(x-1)

Functions Defined in Pieces While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domain. These are called piecewise functions.

Examples: -x ; x < 0 y = x 2 ; 0 < x < 1 1 ; x > 1 -x ; 0 < x < 1 y = 2x – 2 ; 1< x < 2

The Absolute Value Function The absolute value function is defined piecewise:

Composite Functions

Examples f(x) = x 2 + 1g(x) = x- 7 Find: g(f(2)) f(g(2)) g(g(3)) f(f(x)) g(f(x)) g(g(x))

Trig Review Complete Packet (will be part of HW #6) on your own Seek help either during seminar or at next week’s review session

1.3 Exponential Functions

Slide Exponential Growth Exponential Decay Applications The Number e …and why Exponential functions model many growth patterns. What you’ll learn about…

Slide Exponential Function The domain of f(x) = a x is (-∞, ∞) and the range is (0, ∞). Compound interest investment and population growth are examples of exponential growth.

Slide Exponential Growth

Slide Exponential Decay

Slide Exponential Growth and Exponential Decay

Graphing Exponential Functions Graph y = 2 x ◦ x-intercept:_______ ◦ y -intercept:_______ ◦ Domain:_______ ◦ Range: _______ ◦ Type: _______ Slide Graph y = 2 -x x-intercept:_______ y -intercept:_______ Domain:_______ Range: _______ Type: _______

Slide Rules for Exponents See page 21 to review these! Half-life Exponential functions can also model phenomena that produce decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a non-radioactive state by emitting energy in the form of radiation.

Use the Law of Exponents to expand or condense 1. a x a y 2. (a x ) y 3. a x b x 4. (a/b) y Slide 1- 18

Slide Example Exponential Functions [-5, 5], [-10,10]

Rewrite the exponential expression to have the indicated base (9) 2x, base 3 (1/8) 2x, base 2 Slide Applications The Population of Knoxville is 500,000 and is increasing at the rate of 3.75% annually. Approximately when will the population reach 1 million? Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 g present initially. When will there only be 1 g of the substance left?

Interest Problems Simple Interest Formula Compound Interest Formula Interest compounded continuously ◦ How much would you get if P = $1, r = 100% and the principal were compounded continuously (every second of each day for 365 days) for one year? Slide 1- 21

Slide The Number e f(x) = (1 + 1/x) x

Slide The Number e

Slide Example The Number e [0,100] by [0,120] in 10’s

Slide 1- 25

Slide 1- 26

Slide Quick Quiz Sections 1.1 – 1.3

Slide Quick Quiz Sections 1.1 – 1.3

Slide Warm-Up

1.4 Parametric Equations

Slide Relations Lines and Other Curves What you’ll learn about… …and why Parametric equations can be used to obtain graphs of relations and functions.

Slide Relations A relation is a set of ordered pairs (x, y) of real numbers. The graph of a relation is the set of points in a plane that correspond to the ordered pairs of the relation. If x and y are functions of a third variable t, called a parameter, then the equations that define x and y are parametric equations.

Slide Parametric Curve, Parametric Equations Lines, line segments and many other curves can be defined parametrically.

General parametric equations involving angular measure: x = v 0 cos θ t and y = -16t 2 + v 0 sin θ t + s Ex. 1: Consider the path followed by an object that is propelled into the air as an angle of 45 degrees with an initial velocity of 48 ft/sec. The object will follow a parabolic path. Write a Cartesian equation and a set of parametric equations to model this example. Slide 1- 35

Graph each set of parametric equations, then find the Cartesian equation relating the variables (eliminate the parameter): x = 2t + 1 y = 2 – t Cartesian Equation: Slide t012 x y

Slide x = r 2 – 3r + 1 y = r + 1 Cartesian Equation: r x y

x = sin r y = cos r Cartesian Equation: Slide r x y

x = t 3 y = t 2 /2 Slide t x y