Announcements Topics: -section 1.2 (some basic common functions) -section 1.3 (transformations and combinations of functions) -section 1.4 (exponential functions) Homework: read sections 1.2, 1.3, and 1.4 in your textbook work on exercises from the textbook in sections 1.2, 1.3, and 1.4 work on Assignment 1 and Assignment 2

Functions can be described in 4 ways: Numerically (table of values) Geometrically (graph) Algebraically (explicit formula) Verbally (description in words)

Modeling Exercise: Example #20: You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Sketch a rough graph of the temperature of the pie as a function of time. verbal description of a function

Catalogue of Important Functions Download file from website and review + memorize these functions (names, shape of graph, important properties)

Linear Functions slope: point-slope equation: slope-y-intercept equation:

Linear Functions slope: point-slope equation: slope-y-intercept equation:

Linear Model for the Population of Canada Data: YearTime, t Population, P(t) (in thousands)

Linear Model for the Population of Canada Create a linear model for the population of Canada as a function of time using the first two data points.

Linear Model for the Population of Canada Use this model to predict Canada’s population in 2006: Actual observed population in 2006:

Polynomial Functions A polynomial is a function of the form where is a nonnegative integer (0, 1, 2, 3, 4, …) and the numbers are constants called the coefficients of the polynomial. Domain: Degree:

Polynomials Example: Quadratic Function: Complete the square to find the vertex: Domain: Range:

Polynomials Example: Cubic Function: Expand to standard form: Domain: Range: degree: n=3

Polynomials Note 1: Polynomials have nice properties (domain is all real numbers, graphs are smooth and continuous, + more…) and for this reason are used in calculus whenever possible for simple calculations Note 2: A linear function, f(x)=mx+b, is just a polynomial of degree 1.

Power Functions A power function is a function of the form where is a constant. Note: Although can be any real number, we usually omit the case when

Power Functions Some special cases: a=2: Shape: parabola Vertex: (0,0) Domain: Range: ** Graphs of look similar

Power Functions Some special cases: a=3: Shape: cubic parabola Domain: Range: **Graphs of look similar

Power Functions Some special cases: a=1/2: Shape: half of a parabola Domain: Range: square root function

Power Functions Some special cases: a=1/3: Shape: cubic parabola Domain: Range: cube root function

Power Functions Some special cases: a=-1: Shape: hyperbola Domain: Asymptotes: Range: rational function

Power Functions Some special cases: a=-2: Shape: hyperbola Domain: Asymptotes: Range: rational function

Rational Functions A rational function f is a ratio of two polynomials: where and are polynomials and Examples:

Algebraic Functions A combination of any of the the previous functions using algebraic operations is called an algebraic function. Example:

on your own: In the text, read pages 31, 32, and 33 to briefly review Trigonometric Functions, Exponential Functions, and Logarithmic Functions

Transformations: Scaling, Reflecting, Shifting -It can be easier to graph a function if we recognize it as a series of transformations of a basic function -Summary of rules on page 37 in text -Use online graphing calculator and/or wolfram alpha website to check your answers

Transformations: Scaling, Reflecting, Shifting Example 1: Graph Reflect in the x-axis (multiply each y-coordinate by -1):

Transformations: Scaling, Reflecting, Shifting Example 2: Graph First, re-write it so we can easily identify transformations: Graph base function:

Transformations: Scaling, Reflecting, Shifting Example 3: Graph Compress horizontally by a factor of 2 Reflect graph in the x-axis (multiply each y-coordinate by -1):

Combinations of Functions Adding/Subtracting Functions The sum of the functions and is the function defined by The difference of the functions and is the function defined by

Combinations of Functions Multiplying/Dividing Functions The product of the functions and is the function defined by The quotient of the functions and is the function defined by

Combinations of Functions Composition of Functions The composition of the functions and is the function defined by Note: is called the outer function is called the inner function

Combinations of Functions Diagram: Note: In general,

Combinations of Functions Example: Use the table of values below to find the values of (a) (b) (c) x f(x) g(x)632123

Exponential Functions An exponential function is a function of the form where is a positive real number called the base and is a variable called the exponent. Domain: Range:

Graphs of Exponential Functions When a>1, the function is increasing. When a<1, the function is decreasing. y=0 is a horizontal asymptote Memorize!!!

Transformation of an Exponential Function Graph Recall: is a special irrational number between 2 and 3 that is commonly used in calculus Approximation:

Laws of Exponents Examples:

Exponential Models P. 54 #24. Suppose you are offered a job that lasts one month. Which of the following methods of payment do you prefer? I.One million dollars at the end of the month. II.One cent on the first day of the month, two cents on the second day, four cents on the third day, and, in general, cents on the nth day.