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##### Chapter 2 – Functions

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