Piecewise Functions Piecewise-Defined Functions. 7/9/2013 Piecewise Defined Functions 2 Piecewise-Defined Functions Driving to Aunt Berthas House Your.

Slides:

Advertisements

Similar presentations

TWO STEP EQUATIONS 1. SOLVE FOR X 2. DO THE ADDITION STEP FIRST

Advertisements

Slide 1 Insert your own content. Slide 2 Insert your own content.

1 Chapter 40 - Physiology and Pathophysiology of Diuretic Action Copyright © 2013 Elsevier Inc. All rights reserved.

Solving Absolute-Value Inequalities

1 When you see… Find the zeros You think…. 2 To find the zeros...

Graph of a Curve Continuity This curve is _____________These curves are _____________ Smoothness This curve is _____________These curves are _____________.

Graph of a Curve Continuity This curve is continuous

Measures of Variation. Median, Quartiles, Inter-Quartile Range and Box Plots. Measures of Spread Remember: The range is the measure of spread that goes.

Coordinate Plane Practice The following presentation provides practice in two skillsThe following presentation provides practice in two skills –Graphing.

The Slope Of A Line Objective: To find the slope of a line and to graph a line given its slope and a point.

DIVIDING INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

ADDING INTEGERS 1. POS. + POS. = POS. 2. NEG. + NEG. = NEG. 3. POS. + NEG. OR NEG. + POS. SUBTRACT TAKE SIGN OF BIGGER ABSOLUTE VALUE.

MULTIPLICATION EQUATIONS 1. SOLVE FOR X 3. WHAT EVER YOU DO TO ONE SIDE YOU HAVE TO DO TO THE OTHER 2. DIVIDE BY THE NUMBER IN FRONT OF THE VARIABLE.

SUBTRACTING INTEGERS 1. CHANGE THE SUBTRACTION SIGN TO ADDITION

MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

Who Wants To Be A Millionaire? Decimal Edition Question 1.

Lecture 11: Algorithms and Time Complexity I Discrete Mathematical Structures: Theory and Applications.

Show Order 2 Shows 8:00 – 10:30 Weigh in 11:00 Ewes – by Weight Champion Ewe Market Classes (Hamp) Champion Drive 1:00 – 2:30 Weigh in 2 nd show (approximate.

Operations Basic Arithmetic Operations. 7/9/2013 Operations 2 2 A Subtraction ( – ) Division ( / ) Notation … addition of negatives … multiplication of.

How to Tame Them How to Tame Them

What is it and how do I know when I see it?

O X Click on Number next to person for a question.

Numbers & Data Basic Concepts. 7/9/2013 Numbers & Data 2 2 Data Uses of Numbers Counting Finding Differences and Totals Averages and Percentages Rates.

Logarithmic Functions

Difference, Product and Quotient Functions

Logarithmic Equations

1 Directed Depth First Search Adjacency Lists A: F G B: A H C: A D D: C F E: C D G F: E: G: : H: B: I: H: F A B C G D E H I.

Take from Ten First Subtraction Strategy -9 Click on a number below to go directly to that type of subtraction problems

Constant, Linear and Non-Linear Constant, Linear and Non-Linear

Warm-Up Pick up: A Giancoli book Far back left cabinet 1 of each paper at the front 3 Equation Sheets 1 Kinematics Multiple Choice 1 Kinematics Free Response.

Chapter 1: Expressions, Equations, & Inequalities

2.5 Using Linear Models Month Temp º F 70 º F 75 º F 78 º F.

Energy & Green Urbanism Markku Lappalainen Aalto University.

Past Tense Probe. Past Tense Probe Past Tense Probe – Practice 1.

Solving Absolute Value Equations Solving Absolute Value Equations

Do Now 4 Find the equation of a line through the points (7, -2) and (3, -1).

Limits (Algebraic) Calculus Fall, What can we do with limits?

DISTANCE: (d=rt).

Addition 1’s to 20.

25 seconds left…...

Test B, 100 Subtraction Facts

By Lia Servidio and Kelly Lively

U2 L5 Quotient Rule QUOTIENT RULE

STEP FUNCTIONS 3.9. INTRODUCTION In 2007, the U.S. postage rate for first class flats was $0.70 for the first ounce plus $0.17 for each additional ounce.

11 = This is the fact family. You say: 8+3=11 and 3+8=11

X y x y Review of definition: A function is a relation for which there is exactly one value of the dependent variable for.

2 4 Theorem:Proof: What shall we do for an undirected graph?

Let’s take a 15 minute break Please be back on time.

Warm UP September 9, 2013 Explain one thing you know about measurement and mass. Next on page 15 of your notebook begin setting up Cornell notes with todays.

O X Click on Number next to person for a question.

Patient Survey Results 2013 Nicki Mott. Patient Survey 2013 Patient Survey conducted by IPOS Mori by posting questionnaires to random patients in the.

5.1 Accumulated Changes Example 1: An objects travels with a velocity of 15 mph. What is the distance traveled after 4 hours t v Distance = area.

Section Functions Evaluate a Function Identify a Function Given By An Equation Find the Domain of a Function Graph of a Function Vertical Line Test.

Graphing Piecewise Functions

Math 2: Unit 5B Day 3 How do we solve problems using piecewise functions, step functions, and greatest integer functions?

Unit 1 – First-Degree Equations and Inequalities

Section 1.7 Piecewise Functions. Vocabulary Absolute Value Function – a piecewise function, written as f(x)=, where f(x) 0 for all values of x. Greatest.

Dear Power point User, This power point will be best viewed as a slideshow. At the top of the page click on slideshow, then click from the beginning.

2-6: Special Functions Direct Variation: A linear function in the form y = kx, where k 0 Constant: A linear function in the form y = b Identity:

2-6 Special Functions Objectives Students will be able to: 1) identify and graph step, constant, and identity functions 2) Identify and graph absolute.

1.7 Piecewise Functions Objective: Identify and graph piecewise functions including greatest integer, step, and absolute value functions.

Increasing, Decreasing, and Piecewise Functions; Applications

2-6 Special Functions.

The velocity is constant and the distance is:

2-6: Absolute Value Functions

The velocity is constant and the distance is:

2-6: Special Functions Direct Variation: A linear function in the form y = kx, where k 0 Constant: A linear function in the form y = b Identity:

Presentation transcript:

Piecewise Functions Piecewise-Defined Functions

7/9/2013 Piecewise Defined Functions 2 Piecewise-Defined Functions Driving to Aunt Berthas House Your family drives to Aunt Berthas house for Thanksgiving dinner, a distance of 100 miles You average 50 mph each way, visit for 5 hours and return home Define and graph D(t) as distance from home at time t

7/9/2013 Piecewise Defined Functions 3 Piecewise-Defined Functions Driving to Aunt Berthas House Define and graph D(t) as distance from home at time t D(t) = 50t, if 0 t < 2 100, if 2 t < 7 –50t + 450, if 7 t 9 Questions: What is the domain of D ? What is the range of D ? Is function D(t) continuous ?

7/9/2013 Piecewise Defined Functions 4 Piecewise-Defined Functions Driving to Aunt Berthas House Slope = D t = –100 miles 2 hours = –50 hr mi D(t) t D(t) = 50t, if 0 t < 2 100, if 2 t < 7 –50t + 450, if 7 t 9 = 100 miles 2 hours = 50 hr mi Slope = D t Domain Range

7/9/2013 Piecewise Defined Functions 5 Piecewise-Defined Functions The Postal Function In 2008 first-class letter postage P(x) for letters of weight x ounces was assigned as follows: P(x) = , if 0 x 1, if 1 < x 2, if 2 < x 3, if 3 < x 3.5, if 3.5 < x , if 4 < x 5 Question: Graph ?

7/9/2013 Piecewise Defined Functions 6 Piecewise-Defined Functions The Postal Function P(x) x Question: Graph ? A step function Domain ? Range ? Question: Is the function P(x) continuous ?

7/9/2013 Piecewise Defined Functions 7 The Greatest-Integer Function x nxnx Piecewise-Defined Functions n x is the greatest integer x For each non-integer x the value of n x is the largest integer to the left of x Between consecutive integers is constant x nxnx x

7/9/2013 Piecewise Defined Functions 8 The Greatest-Integer Function x nxnx Piecewise-Defined Functions n x is the greatest integer x For integer a, if a < x < a + 1, then Questions : Domain Range ? ? nxnx x nxnx x

7/9/2013 Piecewise Defined Functions 9 The Greatest-Integer Function Piecewise-Defined Functions n x is the greatest integer x Questions : Where ? Is continuous ? x nxnx nxnx x = k k x k What is n k for some integer k? =

7/9/2013 Piecewise Defined Functions 10 x A(x) Years Since Birth Current Age Piecewise-Defined Functions How Old Are You ? In your first year, what is your age ? When does your age change ? How old are you during your third year ? 0 years At 1 year 2 years

7/9/2013 Piecewise Defined Functions 11 Piecewise-Defined Functions How Old Are You ? In your first year, what is your age ? In general, if you are n years old, in what year of your life are you ? 0 years Year n+1 x A(x) Years Since Birth Current Age

7/9/2013 Piecewise Defined Functions 12 x F(x) –1 1 The Salt-and-Pepper Function Piecewise-Defined Functions 1 Question: the domain of F(x) ? the range of F(x) ? F(x) = –1, if x rational, if x irrational Slope not defined Nowhere-continuous function F(0), F(-6), F(), F( ), F 2 ? ( ) What are

7/9/2013 Piecewise Defined Functions 13 The Absolute Value Function Definition The absolute value of a number a, written a, is defined by The absolute value function is for all x Piecewise-Defined Functions a = a –a, if a 0, if a < 0 F(x) x 1 2 –112 –2 F(x) = | x | ( 2, 2 ) || ( -2, -2 ) ||

7/9/2013 Piecewise Defined Functions 14 The Absolute Value Function The absolute value function is for all x Piecewise-Defined Functions F(x) = | x | F(x) x 1 2 –112 –2 ( 2, 2 ) || ( -2, -2 ) || Question: What is the domain of F(x) ? What is the range of F(x) ? Is F(x) continuous ? F(0), F(-6), F(), F( ), F 2 ? ( ) What are : YES

7/9/2013 Piecewise Defined Functions 15 Absolute Value Function Variations Example 1 : Vertex: f(-1) = 0 Piecewise-Defined Functions Questions: Domain of f(x) ? Range of f(x) ? Is f(x) continuous ? What are: f(0), f(-6), f(), f(- ) ? … for all x f(x) x 1 2 –112 –2 ( 1, 2 ) || ( -3, -2 ) || f(x) = x + 1 | |

7/9/2013 Piecewise Defined Functions 16 Absolute Value Function Variations Example 2 : Vertex: f(1) = 0 Piecewise-Defined Functions … for all x f(x) = x – 1 | | f(x) x 1 2 –112 –2 ( 3, 2 ) || ( -1, -2 ) || Questions: Domain of f(x) ? Range of f(x) ? Is f(x) continuous ? What are: f(0), f(-6), f(), f(- ) ?

7/9/2013 Piecewise Defined Functions 17 Think about it !