Understanding Functions: Definitions, Graphs, and Transformations

Chapter 2 – Functions 18 Days

Table of Contents • 2.1 Definition of a Function • 2.2 Graphs of Functions • 2.3 Quadratic Functions • 2.4 Operations on Functions • 2.5 Inverse Functions • 2.6 Variation

2.1 Definition of a Function Two Days

Functions* • For the function • Find f(a) • Find f(a-1) • Find • Find

Homework p 148 (# 5,8,14-28 even, 45,47,49,52,57)

Day 2 Def Function, Domain, Range, Increasing/Decreasing, Vert Line Test, Def Linear Function Evaluating (p148 #13)

Definition of a Function • A function F from a set D to a set E is a correspondence that assigns each element x of D to exactly one element y in E.

Domain and Range • Domain – The set D is the domain of the function. Domain is the set of all possible inputs. • Range – The set E is the range of the function. Range is the set of all possible outputs. • The element y in E is the value of f at x also called the image of x under f.

Function Mapping • We say that f maps D into E. • Two functions f and g are equal if and only if f(x) = g(x) for all x in D.

Graphs of Functions • The graph of a function f is the graph of the equation y = f(x) for all x in the domain of f. • The vertical line test can be used to determine if a graph represents a function. • What does the vertical line test represent in terms of a function mapping?

Increasing and Decreasing Functions -f is increasing when f(a)<f(b) and a<b. -f is decreasing when f(b)>f(c) and b<c. -f is constant when f(x)=f(y) for all x and y.

Evaluating Functions • Given • Determine the domain of g. • Evaluate g(-3) • Evaluate

Sketching Functions • What is the difference between sketching and graphing a function? • Why would we sketch a function as opposed to graph a function?

Sketching Functions • Sketch the following functions and determine the domain, range, and intervals of decreasing, increasing, and constant value:

Finding Linear Functions • We can find linear functions in the same way that we find the equation of a line. • If f is a linear function such that f(-3)=6 and f(2)=-12, find f(x) where x is any real number.

Applications Problems • Pg 150 #57, 59

Homework p 148 (# 15,32,34,35,46,48,50,53,54,60,63,65)

2.2 Graphs of Functions Four Days

Parent Functions -Name of Family -Parent Equation -General Equation -Locator Point -Domain -Range

Graph Shifting and Reflections • Parent: • Shift up k units: • Shift down k units: • Shift right h units: • Shift left h units • Combined Shift: • (right h units, up k units)

Graph Shifting and Reflections • Parent: • Reflection in x-axis: • Vertical Stretch a>1 • Vertical Shrink 0<a<1 • Horizontal Stretch 0<c<1 : • Horizontal Compression c>1: • Combined Transformation:

Graph Shifting and Reflections • Graph the following using translations:

Homework Shifts and Reflections WS

Day 2 – Even and Odd functions. Vertical and Horizontal stretching and compressing of graphs.

Even and Odd Functions • f is an even function if f(-x)=f(x) for all x in the domain. • Even functions have symmetry with respect to the y-axis. • Ex: • f is an odd function if f(-x)=-f(x) for all x in the domain. • Odd functions have symmetry with respect to the origin. • Ex:

Family Functions and Shifts • A parent function is the simplest function in a family of certain characteristics. • A translation shifts the graph horizontally, vertically, or both. Resulting in a graph of the same shape in a different location. • A reflection over the x-axis changes y-values to their opposites.

Family Functions and Shifts • A vertical stretch multiplies all y-values by the same factor greater than 1. • A vertical shrink reduces all y-values by the same factor between 0 and 1. • Each member of a family of functions is a transformation, or change, of the parent function. • A horizontal compression divides all x-values by the same factor greater than 1. • A horizontal stretch divides all x-values by the same factor between 0 and 1.

Graph Shifting and Reflections • Parent: • Shift up k units: • Shift down k units: • Shift right h units: • Shift left h units • Combined Shift: • (right h units, up k units)

Graph Shifting and Reflections • Parent: • Reflection in x-axis: • Vertical Stretch a>1 • Vertical Shrink 0<a<1 • Horizontal Stretch 0<c<1 : • Horizontal Compression c>1: • Combined Transformation:

Homework pg 164 (# 2,3,5,7,8,13,15,17,20,31-36,39 a-f, 41,42,45)

Day 3 – Piecewise functions and questions from the previous 2 days. Application of Piecewise functions (pg 168 #66)

Piecewise Functions • Piecewise functions are defined by more than one expression over different intervals. • Absolute Value is actually a piecewise defined function.

Piecewise Functions • Lets graph the following piecewise defined function.

Applications of Piecewise Functions • An electric company charges its customers $0.0577 per kWh for the first 1000kWh, $0.0532 for the next 4000kWh, and $0.0511 for any over 5000kWh. Write a piecewise defined function C for a customer’s bill of x kWhs. • How much will a customer’s bill be if they used 4300kWh of electricity?

Homework pg 167 (# 47-50,53,54,55,56,63-65)

Day 4 – Graphing Piecewise functions WS. Working day for students.

Homework Graphing Piecewise Functions WS

2.3 Quadratic Functions Two Days

Day 1 – Standard form of a quadratic. Vertex form of a quadratic. Completing the square. Finding x and y intercepts.

Quadratic Functions • Standard form of a Quadratic: • Vertex form of a Quadratic:

Competing the Square

Finding x and y intercepts • To find the x-intercept, set y=0. Solve for x. • To find the y-intercept, set x=0. Solve for y. • Find the x and y intercepts of the following:

Homework Vertex and Intercepts WS

Day 2 – Vertex formula and Theorem on Max/Min Values.

Understanding Functions: Definitions, Graphs, and Transformations

Understanding Functions: Definitions, Graphs, and Transformations

Presentation Transcript

Sea Ice

Sea Ice