Chapter 2: Analysis of Graphs of Functions

Slides:



Advertisements
Similar presentations
1.4 – Shifting, Reflecting, and Stretching Graphs
Advertisements

Graphical Transformations
Section 2.5 Transformations of Functions. Overview In this section we study how certain transformations of a function affect its graph. We will specifically.
Functions: Transformations of Graphs Vertical Translation: The graph of f(x) + k appears as the graph of f(x) shifted up k units (k > 0) or down k units.
Using Transformations to Graph Quadratic Functions 5-1
Copyright © 2007 Pearson Education, Inc. Slide 2-1 Chapter 2: Analysis of Graphs of Functions 2.1 Graphs of Basic Functions and Relations; Symmetry 2.2.
Relations and Functions Linear Equations Vocabulary: Relation Domain Range Function Function Notation Evaluating Functions No fractions! No decimals!
Copyright © 2007 Pearson Education, Inc. Slide 2-1.
Copyright © 2011 Pearson Education, Inc. Slide Vertical Stretching Vertical Stretching of the Graph of a Function If a point lies on the graph.
Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 3 Quadratic Functions and Equations.
Apply rules for transformations by graphing absolute value functions.
2.7 Graphing Absolute Value Functions The absolute value function always makes a ‘V’ shape graph.
Function - 2 Meeting 3. Definition of Composition of Functions.
Graphical Transformations. Quick Review What you’ll learn about Transformations Vertical and Horizontal Translations Reflections Across Axes Vertical.
2.7 Absolute Value Tranformations
Copyright © 2007 Pearson Education, Inc. Slide 2-1.
Graphing Quadratic Functions using Transformational Form The Transformational Form of the Quadratic Equations is:
2-7 Absolute Value Function Objective: I can write and graph an absolute value function.
Copyright © 2011 Pearson Education, Inc. Slide
Transformations of Graphs
CHAPTER 2: More on Functions
Transforming Linear Functions
Quadratic Functions and Their Graphs
Lesson 2-6 Families of Functions.
2.6 Families of Functions Learning goals
13 Algebra 1 NOTES Unit 13.
Using Transformations to Graph Quadratic Functions 5-1
Transformations of Graphs
Interesting Fact of the Day
Use Absolute Value Functions and Transformations
2-7 Absolute Value Functions and Graphs
Absolute Value Functions
Graph Absolute Value Functions using Transformations
2.6 Translations and Families of Functions
Functions and Their Graphs
Graphs of Quadratic Functions
Use Absolute Value Functions and Transformations
3.5 Transformations of Graphs
Transformations: Review Transformations
Chapter 2: Analysis of Graphs of Functions
Section 2.5 Transformations.
Graph Absolute Value Functions using Transformations
2.5 Stretching, Shrinking, and Reflecting Graphs
Learning Objectives for Section 2.2
Pre-AP Pre-Calculus Chapter 1, Section 6
Objectives Transform quadratic functions.
Graph Absolute Value Functions using Transformations
Chapter 2: Analysis of Graphs of Functions
Lesson 5-1 Graphing Absolute Value Functions
Chapter 15 Review Quadratic Functions.
Transformations of Graphs
Graph Transformations
Chapter 15 Review Quadratic Functions.
Graphical Transformations
Warm-up: Welcome Ticket
2.7 Graphing Absolute Value Functions
Parent Functions and Transformations
Transformation rules.
1.5b Combining Transformations
2-6 Families of Functions
CHAPTER 2: More on Functions
2.7 Graphing Absolute Value Functions
2.1 Transformations of Quadratic Functions
6.4a Transformations of Exponential Functions
1.5b Combining Transformations
Chapter 2: Analysis of Graphs of Functions
6.4c Transformations of Logarithmic functions
What is the NAME and GENERAL EQUATION for the parent function below?
2.6 transformations of functions Parent Functions and Transformations.
1.3 Combining Transformations
Presentation transcript:

Chapter 2: Analysis of Graphs of Functions 2.1 Graphs of Basic Functions and Relations; Symmetry 2.2 Vertical and Horizontal Shifts of Graphs 2.3 Stretching, Shrinking, and Reflecting Graphs 2.4 Absolute Value Functions 2.5 Piecewise-Defined Functions 2.6 Operations and Composition

2.3 Vertical Stretching Vertical Stretching of the Graph of a Function If a point lies on the graph of then the point lies on the graph of If then the graph of is a vertical stretching of the graph of by applying a factor of c.

2.3 Vertical Shrinking Vertical Shrinking of the Graph of a Function If a point lies on the graph of then the point lies on the graph of If then the graph of is a vertical shrinking of the graph of by applying a factor of c.

2.3 Horizontal Stretching and Shrinking Horizontal Stretching and Shrinking of the Graph of a Function If a point lies on the graph of then the point lies on the graph of If then the graph of is a horizontal stretching of the graph of (b) If then the graph of is a horizontal shrinking of the graph of

2.3 Reflecting Across an Axis Reflecting the Graph of a Function Across an Axis For a function defined by the following are true. (a) the graph of is a reflection of the graph of f across the x-axis. (b) the graph of is a reflection of the graph of f across the y-axis.

2.3 Example of Reflection Given the graph of sketch the graph of (a) (b) Solution (a) (b)

2.3 Combining Transformations of Graphs Example Describe how the graph of can be obtained by transforming the graph of Sketch its graph. Solution Since the basic graph is the vertex of the parabola is shifted right 4 units. Since the coefficient of is –3, the graph is stretched vertically by a factor of 3 and then reflected across the x-axis. The constant +5 indicates the vertex shifts up 5 units. shift 4 units right shift 5 units up vertical stretch by a factor of 3 reflect across the x-axis

Graphs:

2.3 Caution in Translations of Graphs The order in which transformations are made can be important. Changing the order of a stretch and shift can result in a different equation and graph. For example, the graph of is obtained by first stretching the graph of by a factor of 2, and then translating 3 units upward. The graph of is obtained by first translating horizontally 3 units to the left, and then stretching by a factor of 2.

2.3 Transformations on a Calculator- Generated Graph Example The figures show two views of the graph and another graph illustrating a combination of transformations. Find the equation of the transformed graph. Solution The first view indicates the lowest point is (3,–2), a shift 3 units to the right and 2 units down. The second view shows the point (4,1) on the graph of the transformation. Thus, the slope of the ray is Thus, the equation of the transformed graph is First View Second View