Types of Functions.

Slides:

Advertisements

Similar presentations

How do we perform transformations of functions?

Advertisements

Essential Question: In the equation g(x) = c[a(x-b)] + d what do each of the letters do to the graph?

Lesson 1-3 New Functions from Old Functions Part 1 - Part 1 tbn3.gstatic.com/images?q=tbn:ANd9GcTMSNbfIIP8t1Gulp87xLpqX92qAZ_vZwe4Q.

Aim: How do transformations affect the equations and graphs of functions? Do Now: Graph y = -.25x2 – 4 and describe some of the important features. Axis.

FUNCTIONS AND MODELS New Functions from Old Functions In this section, we will learn: How to obtain new functions from old functions and how to.

Objective Transform polynomial functions..

The vertex of the parabola is at (h, k).

Essential Question: In the equation f(x) = a(x-h) + k what do each of the letters do to the graph?

CN College Algebra Ch. 2 Functions and Their Graphs 2.5: Graphing Techniques: Transformations Goals: Graph functions using horizontal and vertical shifts.

Table of Contents Functions: Transformations of Graphs Vertical Translation: The graph of f(x) + k appears.

Transforming reciprocal functions. DO NOW Assignment #59  Pg. 503, #11-17 odd.

B. Functions Calculus Introduction A relation is simply a set of ordered pairs. A function is a set of ordered pairs in which each x-value is paired.

1 The graphs of many functions are transformations of the graphs of very basic functions. The graph of y = –x 2 is the reflection of the graph of y = x.

Quadratic functions are defined by: y = f(x) = ax 2 +bx + c = 0 The graph of a quadratic function is a parabola. The most basic quadratic function is:

Graphing Quadratics.

College Algebra 2.7 Transformations.

Transformation of Functions Recognize graphs of common functions Use shifts to graph functions Use reflections to graph functions Use stretching & shrinking.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 3 Quadratic Functions and Equations.

Transform quadratic functions.

2.6 – Transformations of Graphs

1.6 PreCalculus Parent Functions Graphing Techniques.

5.1 Stretching/Reflecting Quadratic Relations

6-8 Graphing Radical Functions

3-8 transforming polynomial functions

2.7 Graphing Absolute Value Functions The absolute value function always makes a ‘V’ shape graph.

Lecture # 3 MTH 104 Calculus and Analytical Geometry.

Warm up Determine whether the graph of each equation is symmetric wrt the x-axis, the y-axis, the line y = x the line y = -x or none x-axis, y-axis,

1.6 Transformation of Functions

Sullivan PreCalculus Section 2.5 Graphing Techniques: Transformations

3.4 Graphing Techniques; Transformations. (0, 0) (1, 1) (2, 4) (0, 2) (1, 3) (2, 6)

3-2 Families of Graphs Pre Calc A. Parent Graphs.

Chapter 2 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc Transformations of Functions.

1.3 New Functions from Old Functions In this section, we will learn: How to obtain new functions from old functions and how to combine pairs of functions.

3.4 Graphs and Transformations

 Let c be a positive real number. Vertical and Horizontal Shifts in the graph of y = f(x) are represented as follows. 1. Vertical shift c upward:

TRANSFORMATION OF FUNCTIONS FUNCTIONS. REMEMBER Identify the functions of the graphs below: f(x) = x f(x) = x 2 f(x) = |x|f(x) = Quadratic Absolute Value.

Chapter 3 Non-Linear Functions and Applications Section 3.2.

 How would you sketch the following graph? ◦ y = 2(x – 3) 2 – 8  You need to perform transformations to the graph of y = x 2  Take it one step at a.

To remember the difference between vertical and horizontal translations, think: “Add to y, go high.” “Add to x, go left.” Helpful Hint.

Parent Functions and Transformations. Transformation of Functions Recognize graphs of common functions Use shifts to graph functions Use reflections to.

Section 3.5 Graphing Techniques: Transformations.

Today in Precalculus Need a calculator Go over homework Notes: Rigid Graphical Transformations Homework.

Chapter 2 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc Transformations of Functions.

2.5 Shifting, Reflecting, and Stretching Graphs. Shifting Graphs Digital Lesson.

2.7 – Use of Absolute Value Functions and Transformations.

Section 1.4 Transformations and Operations on Functions.

 Determine whether the graph of each equation is symmetric wrt the x-axis, the y-axis, the line y = x the line y = -x or none.  1.  2. Warm up.

Entry Task. 2.6 Families of Functions Learning Target: I can analyze and describe transformations of functions Success Criteria: I can describe the effect.

Transformations of Functions. The vertex of the parabola is at (h, k).

Section 2.5 Transformations Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

RECAP Functions and their Graphs. 1 Transformations of Functions For example: y = a |bx – c| + d.

Ch. 1 – Functions and Their Graphs 1.4 – Shifting, Reflecting, and Sketching Graphs.

Warm-Up Evaluate each expression for x = -2. 1) (x – 6) 2 4 minutes 2) x ) 7x 2 4) (7x) 2 5) -x 2 6) (-x) 2 7) -3x ) -(3x – 1) 2.

5-3 Using Transformations to Graph Quadratic Functions.

Warm Up 1. State whether the following functions are even, odd, or neither: a. f(x) = –3x b. f(x) = 2x 3 – 4x 1. State the intervals in which the.

The following are what we call The Parent Functions.

Shifting, Reflecting, & Stretching Graphs 1.4. Six Most Commonly Used Functions in Algebra Constant f(x) = c Identity f(x) = x Absolute Value f(x) = |x|

P.3 Functions & Their Graphs 1.Use function notation to represent and evaluate a function. 2.Find the domain and range of a function. 3.Sketch the graph.

1.2 Mathematical Models: A Catalog of Essential Functions

2.6 Families of Functions Learning goals

Transformation of Functions

Absolute Value Transformations

I can Shift, Reflect, and Stretch Graphs

In this section, we will study the following topics:

Transformation of Functions

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

15 – Transformations of Functions Calculator Required

Chapter 3: Functions and Graphs Section 3

Presentation transcript:

Types of Functions

Type 1: Constant Function f(x) = c Example: f(x) = 1

Type 2: Power Function f(x) = xa

If a is a (+) integer f(x) = xn where n = 1,2,3,4,5….. -Shape depends on if n is even or odd -As n increases the graph becomes flatter near 0 and steeper where x ≥ 1

If a is -1 f(x) = x-1 = 1/x Hyperbola

If a = 1/n Root Function f(x) = x1/n = n√(x)

Polynomial f(x) = axn + bxn – 1 + cx n – 2 …….. Degree (n) – highest exponent value 1st Degree: f(x) = ax + b

2nd Degree: Quadratic: f(x) = ax2+ bx + c Parabola

Higher Degrees

Type 3: Algebraic Functions Can be constructed using algebraic operations (add, subtract, multiplication, division, square root) f(x) = √(x2 + 1) f(x) = x4 – 16x2 + (x-2)3√(x) x + √(x) Shapes vary

Type 4: Trigonometric Functions

Tan(x) = sin(x)/ cos(x)

Type 5: Exponential Functions f(x) = ax

Type 6: Log Function f(x) = logax Inverse exponential

Related Functions By applying certain transformations to graphs of given functions, we can obtain the graphs of related functions

Translations - Shifts Vertical shifts y = f(x) + c shifts c units up shifts c units down

Horizontal shifts y = f(x – c) shifts right c units y = f(x + c) shifts left c units

Stretching and Compressing y = cf(x) stretched vertically by a factor of c y = 1/c f(x) compressed vertically by a factor of c y = f(cx) compressed horizontally by a factor of c y = f(x/c) strectched horizontally by a factor of c

Reflecting y = -f(x) graph reflects about the x-axis y = f(-x) graph reflects about the y-axis

Examples Given y = √(x), sketch a) y = √(x) - 2 b) y = √(x - 2) c) y = - √(x) d) y = 2√(x) e) y = √(-x)