Chapter 3: Functions and Graphs 3.2: Graphs of Functions Essential Question: What can you look for in a graph to determine if the graph represents a function?

Slides:



Advertisements
Similar presentations
Each part of graph is described as: 1)Increasing : function values increase from left to right 2)Decreasing: function values decrease 3)Constant function.
Advertisements

Relationship between First Derivative, Second Derivative and the Shape of a Graph 3.3.
1 Concavity and the Second Derivative Test Section 3.4.
More on Functions and Their Graphs Section 1.3. Objectives Calculate and simplify the difference quotient for a given function. Calculate a function value.
Copyright © Cengage Learning. All rights reserved.
1.3 Graphs of Functions Pre-Calculus. Home on the Range What kind of "range" are we talking about? What kind of "range" are we talking about? What does.
THE ABSOLUTE VALUE FUNCTION. Properties of The Absolute Value Function Vertex (2, 0) f (x)=|x -2| +0 vertex (x,y) = (-(-2), 0) Maximum or Minimum? a =
OBJECTIVES: 1. DETERMINE WHETHER A GRAPH REPRESENTS A FUNCTION. 2. ANALYZE GRAPHS TO DETERMINE DOMAIN AND RANGE, LOCAL MAXIMA AND MINIMA, INFLECTION POINTS,
Graphs of Functions Lesson 3.
1.5 Increasing/Decreasing; Max/min Tues Sept 16 Do Now Graph f(x) = x^2 - 9.
Properties of a Function’s Graph
Section 3.6 – Curve Sketching. Guidelines for sketching a Curve The following checklist is intended as a guide to sketching a curve by hand without a.
Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.
Graphs of Functions Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 x y 4 -4 The domain of the function y = f (x)
Copyright © Cengage Learning. All rights reserved.
Lesson 1.3 Read: Pages Page 38: #1-49 (EOO), #61-85 (EOO)
Lesson 13 Graphing linear equations. Graphing equations in 2 variables 1) Construct a table of values. Choose a reasonable value for x and solve the.
REVIEW Reminder: Domain Restrictions For FRACTIONS: n No zero in denominator! For EVEN ROOTS: n No negative under radical!
Sections 3.3 – 3.6 Functions : Major types, Graphing, Transformations, Mathematical Models.
A function from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is called the.
Chapter Relations & Functions 1.2 Composition of Functions
College Algebra Acosta/Karwowski. CHAPTER 3 Nonlinear functions.
Functions and Their Graphs Advanced Math Chapter 2.
Chapter 5 Graphing and Optimization
Increasing / Decreasing Test
Curve Sketching Lesson 5.4. Motivation Graphing calculators decrease the importance of curve sketching So why a lesson on curve sketching? A calculator.
PreCalculus Sec. 1.3 Graphs of Functions. The Vertical Line Test for Functions If any vertical line intersects a graph in more than one point, the graph.
Graphs of Functions. Text Example SolutionThe graph of f (x) = x is, by definition, the graph of y = x We begin by setting up a partial table.
Relations And Functions. A relation from non empty set A to a non empty set B is a subset of cartesian product of A x B. This is a relation The domain.
2.3 Analyzing Graphs of Functions. Graph of a Function set of ordered pairs.
Copyright © Cengage Learning. All rights reserved. 1 Functions and Their Graphs.
Chapter 3: Functions and Graphs 3.1: Functions Essential Question: How are functions different from relations that are not functions?
Math II Unit 5 (Part 1). Solve absolute value equations and inequalities analytically, graphically and by using appropriate technology.
AIM: WHAT IS A PIECEWISE DEFINED FUNCTION? HW P. 25 #37, 39, 95, 97, P. 39#43, 48 1 EX) Sketch the Piecewise-Defined Function BY HAND x2x+3y 12(1)+35 02(0)+32.
Chapter 2 Linear Relations and Functions BY: FRANKLIN KILBURN HONORS ALGEBRA 2.
Warm-up Determine the equation of this absolute value function. Then, give the intervals of increase and decrease and the domain and range.
(MTH 250) Lecture 2 Calculus. Previous Lecture’s Summary Introduction. Purpose of Calculus. Axioms of Order. Absolute value. Archimedean Property. Axioms.
Trig/Pre-Calculus Opening Activity
Date: 1.2 Functions And Their Properties A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain.
Ch 2 Quarter TEST Review RELATION A correspondence between 2 sets …say you have a set x and a set y, then… x corresponds to y y depends on x x is the.
Domain and Range: Graph Domain- Look horizontally: What x-values are contained in the graph? That’s your domain! Range- Look vertically: What y-values.
Lesson: Objectives: 5.1 Solving Quadratic Equations - Graphing  DESCRIBE the Elements of the GRAPH of a Quadratic Equation  DETERMINE a Standard Approach.
Piecewise Functions Pieces of 2 or more relations Final graph is a function (passes vert. line test) CALCULATOR: – Y= (function)(restriction) – Restriction:
Functions 2 Copyright © Cengage Learning. All rights reserved.
Evaluating Piecewise and Step Functions. Evaluating Piecewise Functions Piecewise functions are functions defined by at least two equations, each of which.
Functions from a Calculus Perspective
Graphs of Polynomial Functions A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate.
Chapter 2: Analysis of Graphs of Functions
Chapter 12 Graphing and Optimization
Chapter 3: Functions and Graphs 3.2: Graphs of Functions
Functions and Their Graphs
1.3 Graphs of Functions Pre-Calculus.
Chapter 2 Applications of the Derivative
Second Derivative Test
How can I analyze graphs of FUNctions?
Curve Sketching Lesson 5.4.
College Algebra Chapter 2 Functions and Graphs
6-5 Linear Inequalities.
Functions and Their Graphs
2-6 Special Functions.
Copyright © Cengage Learning. All rights reserved.
Relationship between First Derivative, Second Derivative and the Shape of a Graph 3.3.
Functions Defined by Graphs
Ch 1.3: Graphs of functions
Analyzing the Graphs of Functions
Write each using Interval Notation. Write the domain of each function.
Graphing linear equations
Chapter 4 Graphing and Optimization
Relationship between First Derivative, Second Derivative and the Shape of a Graph 3.3.
Presentation transcript:

Chapter 3: Functions and Graphs 3.2: Graphs of Functions Essential Question: What can you look for in a graph to determine if the graph represents a function?

3.2: Graphs of Functions Ex 1: Functions Defined by Graphs ▫A graph may be used to define a function or relation. Suppose that the graph below defines a function f. ▫Find:  f (0)  f (3)  f (2)  The domain of f   The range of f  f (0) = 7 f (3) = 0 f (2) = undefined [-8, 2) and (2, 7] [-9, 8]

3.2: Graphs of Functions Ex 2: The Vertical Line Test ▫A graph in a coordinate plane represents a function if and only if no vertical line intersects the graph more than once. ▫ Not a Function Function |

3.2: Graphs of Functions Ex 3: Where a Function is Increasing/Decreasing ▫A function is said to be increasing on an interval if its graph always rises as you move left to right. ▫It is decreasing if its graph always falls as you move left to right ▫A function is said to be constant on an interval if its graph is a horizontal line over the interval

3.2: Graphs of Functions Ex 3: Where a Function is Increasing/Decreasing ▫On what interval is the function f (x) = |x| + |x – 2| increasing? Decreasing? Constant?  Graph the function  It suggests that f is  Decreasing from (- ∞, 0) Increasing on (2, ∞) Constant on [0, 2]

3.2: Graphs of Functions Assignment ▫Page 160  1 – 14, 17 & 18 (all problems)

Chapter 3: Functions and Graphs 3.2: Graphs of Functions Day 2 Essential Question: What can you look for in a graph to determine if the graph represents a function?

3.2: Graphs of Functions Ex 4: Finding Local Maxima and Minima ▫A graph of a function may include some peaks and valleys. ▫The Peak may not be the highest point, but it is the highest point in its area (called a local maximum) ▫A valley may not be the lowest point, but it is the lowest point in its area (called a local minimum) ▫Calculus is usually needed to find exact local maxima and minima. However, they can be approximated with a calculator.

3.2: Graphs of Functions Ex 4: Finding Local Maxima and Minima ▫Graph f (x) = x 3 – 3.8x 2 + x + 1 and find all local maxima and minima.  Graph is shown on calculator  You can find local maxima and minima by using the FMIN and FMAX just like finding the root from a graph.  [Graph] → [more] → [math] → [fmin]/[fmax]

3.2: Graphs of Functions Ex 5: Analyzing a Graph ▫Concavity and Inflection Points  A point where the curve changes concavity is called an inflection point  An inflection point will be always be between a local maximum and local minimum’s x-values  Concavity is used to describe the way a curve bends  Connect two points on a curve, between inflection points  If the line is above the curve, it’s concave up  If the line is below the curve, it’s concave down ▫Open up = concave up, open down = concave down

3.2: Graphs of Functions Ex 5: Analyzing a Graph ▫Graph the function f (x) = -2x 3 + 6x 2 – x + 3 ▫Find  All local maxima and minima of the function  Intervals where the function is increasing/decreasing  All inflection points of the function  Intervals where the function is concave up and where it is concave down

3.2: Graphs of Functions Assignment ▫Page 161  19-27, (odd problems)  Hint #1: Do problems 23 – 27 before 19 & 21  Hint #2: For 33 – 35, find the inflection point first  Hint #3: For 37 & 39: ▫I don’t need to see your graph (part “a”) ▫Find part “c” before part “b” ▫Find part “e” before part “d”

Chapter 3: Functions and Graphs 3.2: Graphs of Functions Day 3 Essential Question: What can you look for in a graph to determine if the graph represents a function?

3.2 Graphs of Functions Ex 6: Graphing a Piecewise Function ▫To graph a piecewise function by hand  Sketch (lightly) each of the graphs  Use the individual domain rule to only use the specified part of the graph & put them together ▫To graph a piecewise function on the calculator  Enter the function in normally  Divide it by the domain of its piece  Inequality symbols are in the test menu (2 nd, 2)  Compound inequalities must be split up

3.2: Graphs of Functions Ex 6: Graphing a Piecewise Function (calculator) ▫Graph ▫On the graphing calculator:  x 2 /(x<1)  x+2/((1<x)(x<4))

3.2: Graphs of Functions Ex 7: The Absolute-Value Function ▫Graph f (x)=|x| ▫This is also a piecewise function ▫  For the second equation, flip the sign on all terms that were inside the absolute value signs.  Domain is split where the stuff inside the absolute value would equal 0 (the x-coordinate of the vertex of the absolute value function)

3.2: Graphs of Functions Ex 7: The Absolute-Value Function #2 ▫Graph f (x)=|2x – 6| + 4  What are the two equations?   Where do the equations split? (Where’s the vertex?)  2x – = 2x x = -2x +10 2x – 6 = x = 6 x = 3, x > 3, x < 3

3.2: Graphs of Functions Ex 8: The Greatest Integer Function ▫Graph f (x)=[x]  We enter the function in as “int x”  Doesn’t look quite right, does it?  To change graphing type  (Only necessary for the greatest integer function)  On the screen to enter functions, press more  Press F3 for “Style”, use the (dot display) setting

3.2: Graphs of Functions Assignment ▫Page 161  (odd problems)

Chapter 3: Functions and Graphs 3.2: Graphs of Functions Uncovered This Year Essential Question: What can you look for in a graph to determine if the graph represents a function?

3.2: Graphs of Functions Ex 9: Parametric Graphing ▫In parametric graphing, both the x and y coordinate are given functions to a 3 rd variable, t. ▫Graph the curve given by  x=2t + 1  y = t 2 – 3 ▫Solution, make a table of values, and sketch

3.2: Graphs of Functions Ex 9: Parametric Graphing ▫x=2t + 1 ▫y = t 2 – 3 ▫Now graph tx = 2t + 1y = t 2 - 3(x, y) -2-31(-3, 1) -2(-1, -2) 01-3(1, -3) 13-2(3, -2) 251(5, 1) 376(7, 6)

3.2: Graphs of Functions Ex 10: Graphing (w/ calc) in parametric mode ▫Change mode (2 nd, mode) to “Param” (5 th down) ▫Now when you go to graph, y(x) is changed to E(t)  You also now enter in two functions at a time (x & y)  To graph y = f (x) in parametric mode  Let x = t and y = f (t)  To graph x = f(y) in parametric mode  Let y = t and x = f (t)  Alter your window  Change the t-step = 0.1

3.2: Graphs of Function Ex 10: Graphing in Parametric Mode 1)Graph ▫Let x = t and 2)Graph x = y 2 – 3y + 1 ▫Let y = t and x = t 2 – 3t + 1