L AST MATHEMATICIAN STANDING Review 1.4-1.6. P ROBLEM 1 Find f(g(x)) and f(f(x)).

Slides:

Advertisements

Similar presentations

1 Example 3 (a) Find the range of f(x) = 1/x with domain [1,  ). Solution The function f is decreasing on the interval [1,  ) from its largest value.

Advertisements

Unit 3 Functions (Linear and Exponentials)

Unit 3 Functions (Linear and Exponentials)

2.3 Stretching, Shrinking, and Reflecting Graphs

Transformation of Functions Section 1.6. Objectives Describe a transformed function given by an equation in words. Given a transformed common function,

Section 1.6 Transformation of Functions

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

Exponential Functions and Their Graphs Section 3-1.

Section 3.2 Notes Writing the equation of a function given the transformations to a parent function.

6.5 - Graphing Square Root and Cube Root

Transformations xf(x) Domain: Range:. Transformations Vertical Shifts (or Slides) moves the graph of f(x) up k units. (add k to all of the y-values) moves.

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.

College Algebra 2.7 Transformations.

Shifting Graphs Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graphs of many functions are transformations.

Copyright © 2009 Pearson Education, Inc. CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra.

1 PRECALCULUS I Dr. Claude S. Moore Danville Community College Composite and Inverse Functions Translation, combination, composite Inverse, vertical/horizontal.

Functions Review.

 Reflections in the coordinate axes of the graph of y = f(x) are represented by: 1. Reflection in the x-axis: h(x) = -f(x) 2. Reflection in the y-axis:

3-8 transforming polynomial functions

Mrs. McConaughyHonors Algebra 21 Graphing Logarithmic Functions During this lesson, you will:  Write an equation for the inverse of an exponential or.

Graphing Reciprocal Functions

Exponential Functions and Their Graphs 2 The exponential function f with base a is defined by f(x) = a x where a > 0, a  1, and x is any real number.

8-2 Properties of Exponential Functions. The function f(x) = b x is the parent of a family of exponential functions for each value of b. The factor a.

Translations and Combinations Algebra 5/Trigonometry.

Function - 2 Meeting 3. Definition of Composition of Functions.

© The Visual Classroom 3.7 Transformation of Functions Given y = f(x), we will investigate the function y = af [k(x – p)] + q for different values of a,

1. 3x + 2 = ½ x – 5 2. |3x + 2| > x – 5 < -3x |x + 2| < 15 Algebra II 1.

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

2.5 Transformations and Combinations of Functions.

Section 1.3 New Functions from Old. Plot f(x) = x 2 – 3 and g(x) = x 2 – 6x + 1 on the same set of axes –What is the relationship between the two graphs?

 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:

Graphing Rational Functions Through Transformations.

Transformations of Functions. Graphs of Common Functions See Table 1.4, pg 184. Characteristics of Functions: 1.Domain 2.Range 3.Intervals where its increasing,

5.4 Logarithmic Functions. Quiz What’s the domain of f(x) = log x?

Section 3.5 Graphing Techniques: Transformations.

Exponential & Logarithmic functions. Exponential Functions y= a x ; 1 ≠ a > 0,that’s a is a positive fraction or a number greater than 1 Case(1): a >

Digital Lesson Shifting Graphs.

Sec 2.4 Transformation of Graphs. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graphs of many functions are transformations.

Quick Crisp Review Graphing a piecewise function Determine relative max and min Graphing a step function 35)a) oddb) even (-3/2, 4)

FUNCTION TRANSLATIONS ADV151 TRANSLATION: a slide to a new horizontal or vertical position (or both) on a graph. f(x) = x f(x) = (x – h) Parent function.

Shifting Graphs. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. As you saw with the Nspires, the graphs of many functions are transformations.

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

Transformation of Functions Sec. 1.7 Objective You will learn how to identify and graph transformations.

Review of Transformations and Graphing Absolute Value

Vocabulary A nonlinear function that can be written in the standard form Cubic Function 3.1Graph Cubic Functions A function where f (  x) =  f (x).

(a) (b) (c) (d) Warm Up: Show YOUR work!. Warm Up.

Vocabulary The distance to 0 on the number line. Absolute value 1.9Graph Absolute Value Functions Transformations of the parent function f (x) = |x|.

I. There are 4 basic transformations for a function f(x). y = A f (Bx + C) + D A) f(x) + D (moves the graph + ↑ and – ↓) B) A f(x) 1) If | A | > 1 then.

1. g(x) = -x g(x) = x 2 – 2 3. g(x)= 2 – 0.2x 4. g(x) = 2|x| – 2 5. g(x) = 2.2(x+ 2) 2 Algebra II 1.

X g(x) y of g f(x) y of f inputoutputinputoutput Domain: Have to make sure that the output of g(x) = - 3. Find.

Section 1.4 Transformations and Operations on Functions.

1 PRECALCULUS Section 1.6 Graphical Transformations.

Exponential & Logarithmic functions. Exponential Functions y= a x ; 1 ≠ a > 0,that’s a is a positive fraction or a number greater than 1 Case(1): a >

Algebra 2 5-R Unit 5 – Quadratics Review Problems.

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

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

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.

Section 1-5 Graphical Transformations. Section 1-5 vertical and horizontal translations vertical and horizontal translations reflections across the axes.

© The Visual Classroom 3.7 Transformation of Functions Given y = f(x), we will investigate the function y = af [k(x – p)] + q for different values of a,

Transforming Linear Functions

Calculus P - Review Review Problems

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|

REVIEW. A. All real numbers B. All real numbers, x ≠ -5 and x ≠ -2 C. All real numbers, x ≠ 2 D. All real numbers, x ≠ 5 and x ≠ 2.

Transforming Linear Functions

Who Wants to Be a Transformation Millionaire?

Graphing Exponential Functions

Warm-up: Graph y =|x| y = x2 y =

6.4a Transformations of Exponential Functions

Transformations.

Transformation of Functions

Presentation transcript:

L AST MATHEMATICIAN STANDING Review

P ROBLEM 1 Find f(g(x)) and f(f(x)).

P ROBLEM 1 - ANSWERS Find f(g(x)) and f(f(x)).

P ROBLEM 2 Find f(g(x)) and its domain.

P ROBLEM 2 Find f(g(x)) and its domain. Domain of f(g(x)) is ( 2, ∞ )

P ROBLEM 3 f(x) = g(x) =

P ROBLEM 3 f(x) = g(x) =

P ROBLEM 4 g(x) = f(x) =

P ROBLEM 4 g(x) = f(x) =

P ROBLEM 5 Find the inverse of h(x).

P ROBLEM 5 Find the inverse of h(x).

P ROBLEM 6 Find g(f(-1)).

P ROBLEM 6 Find g(f(-1)).

For the remaining problems, give the basic function and all the vertical and horizontal transformations in the correct order! Be sure to mention where the horizontal and vertical asymptotes are.

P ROBLEM 7

Basic Function is Horizontal SHIFT right 6 Horizontal shrink BAFO 1/3rd

P ROBLEM 8

Basic Function is Horizontal SHIFT right 4 Vertical SHIFT up 3

P ROBLEM 9

Basic Function is Vertical STRECTH BAFO 2 Vertical SHIFT up 7 HA is at y = 7

P ROBLEM 10

Basic Function is Reflection over the x-axis Vertical SHIFT up 6 Horizontal SHRINK BAFO 1/4 th VA is at X=0

P ROBLEM 11

Basic Function is Reflection over the y-axis Horizontal SHRINK BAFO 1/2 Vertical SHIFT up 1

P ROBLEM 12

Basic Function is Horizontal SHIFT left 5 Horizontal SHRINK BAFO 1/3rd Reflection over the y-axis

P ROBLEM 13

Basic Function is Horizontal STRETCH BAFO 3 Vertical STRETCH BAFO 2 Reflection over the x-axis Vertical SHIFT down 4

P ROBLEM 14

Basic Function is Reflection over the y-axis Vertical SHRINK BAFO 1/2 Vertical SHIFT down 7

P ROBLEM 15

Basic Function is Horizontal SHIFT right 4 Horizontal STRETCH BAFO 2 Vertical SHIFT up 2

P ROBLEM 16 Find and its domain.

P ROBLEM 16 Find and its domain. Domain of inverse is [ 0, ∞ )