P. 82. We will find that the “shifting technique” applies to ALL functions. If the addition or subtraction occurs prior to the function occurring, then.

Slides:

Advertisements

Similar presentations

Session 10 Agenda: Questions from ? 5.4 – Polynomial Functions

Advertisements

Section 8.6 Quadratic Functions & Graphs  Graphing Parabolas f(x)=ax 2 f(x)=ax 2 +k f(x)=a(x–h) 2 f(x)=a(x–h) 2 +k  Finding the Vertex and Axis of Symmetry.

1 Learning Objectives for Section 2.2 You will become familiar with some elementary functions. You will be able to transform functions using vertical and.

Graphs of Radical Functions

THE GRAPH OF A QUADRATIC FUNCTION

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.

Function Families Lesson 1-5.

1.5 Transformations of Some Basic Curves 1 In previous sections, we have graphed equations such as f(x)=x 2 +3 by either translating the basic function.

Unit 3 Functions (Linear and Exponentials)

Unit 3 Functions (Linear and Exponentials)

Objective Transform polynomial functions..

Solving Quadratic Equations by Graphing

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

1 Example 1 (a) Let f be the rule which assigns each number to its square. Solution The rule f is given by the formula f(x) = x 2 for all numbers x. Hence.

Math I, Sections 2.5 – 2.9 Factoring Polynomials

Lesson 4.5. The graph of the square root function,, is another parent function that you can use to illustrate transformations. From the graphs below,

3 Polynomial and Rational Functions © 2008 Pearson Addison-Wesley. All rights reserved Sections 3.1–3.4.

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.

Copyright © Cengage Learning. All rights reserved.

Sections 3.3 – 3.6 Functions : Major types, Graphing, Transformations, Mathematical Models.

Vertical and horizontal shifts If f is the function y = f(x) = x 2, then we can plot points and draw its graph as: If we add 1 (outside change) to f(x),

Homework: p , 17-25, 45-47, 67-73, all odd!

Parent Functions and Transformations. Parent Graphs In the previous lesson you discussed the following functions: Linear Quadratic Cubic, Square root.

Polynomials and rational functions are smaller groups of Algebraic Functions Another group of Algebraic Functions are Rational Power Functions. A rational.

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.

Special Functions and Graphs Algebra II …………… Sections 2.7 and 2.8.

Transforming Polynomial Functions. Recall Vertex Form of Quadratic Functions: a h K The same is true for any polynomial function of degree “n”!

Section 1.2 Analyzing Graphs x is the distance from the y-axis f(x) is the distance from the x-axis p. 79 Figure 1.6 (open dot means the graph does not.

Functions and Models 1. New Functions from Old Functions 1.3.

Sections 11.6 – 11.8 Quadratic Functions and Their Graphs.

Characteristics of Quadratics

2.5 Transformations of Functions What are the basic function formulas we have covered and what does each graph look like? What kinds of ways can we manipulate.

1.6 Transformation of Functions

Ch 6 - Graphing Day 1 - Section 6.1. Quadratics and Absolute Values parent function: y = x 2 y = a(x - h) 2 + k vertex (h, k) a describes the steepness.

FST Quick Check Up Sketch an example of each function: Identity Function Absolute Value Function Square Root Function Quadratic Function Cubic Function.

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,

Quadratic Functions Algebra III, Sec. 2.1 Objective You will learn how to sketch and analyze graph of functions.

 Determine the value of k for which the expression can be factored using a special product pattern: x 3 + 6x 2 + kx + 8  The “x” = x, and the “y” = 2.

Friday, March 21, 2013 Do Now: factor each polynomial 1)2)3)

2nd Level Difference Test means quadratic

Polynomials have the property that the sum, difference and product of polynomials always produce another polynomial. In this chapter we will be studying.

1 Copyright © Cengage Learning. All rights reserved. 2 Polynomial and Rational Functions.

Objectives Vertical Shifts Up and Down

Math 20-1 Chapter 7 Absolute Value and Reciprocal Functions 7.2 Absolute Value Function Teacher Notes.

The absolute-value parent function is composed of two linear pieces, one with a slope of –1 and one with a slope of 1. In Lesson 2-6, you transformed linear.

Warm Up Give the coordinates of each transformation of (2, –3). 4. reflection across the y-axis (–2, –3) 5. f(x) = 3(x + 5) – 1 6. f(x) = x 2 + 4x Evaluate.

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.

1 OCF Reciprocals of Quadratic Functions MCR3U - Santowski.

Copyright © Cengage Learning. All rights reserved. Functions and Graphs 3.

Objective: SWBAT review graphing techniques of stretching & shrinking, reflecting, symmetry and translations.

Rational Functions and Asymptotes Section 2.6. Objectives Find the domain of rational functions. Find horizontal and vertical asymptotes of graphs of.

UNIT 5 REVIEW. “MUST HAVE" NOTES!!!. You can also graph quadratic functions by applying transformations to the parent function f(x) = x 2. Transforming.

Review Chapter 1 Functions and Their Graphs. Lines in the Plane Section 1-1.

Functions from a Calculus Perspective

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

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

Section 1-4 Shifting, Reflecting, and Stretching Graphs

Radical Functions.

Section P.3 Transformation of Functions. The Constant Function.

Unit 2 – Quadratic Functions & Equations. A quadratic function can be written in the form f(x) = ax 2 + bx + c where a, b, and c are real numbers and.

Graph of a Function Ex. Using the graph, find: a)domain b)range c) f (-1), f (1), and f (2)

Absolute Value Function

I can Shift, Reflect, and Stretch Graphs

Graphing Square Root Functions

“Graphing Square Root Functions”

GRAPHING PARABOLAS To graph a parabola you need : a) the vertex

Presentation transcript:

p. 82

We will find that the “shifting technique” applies to ALL functions. If the addition or subtraction occurs prior to the function occurring, then it is a horizontal shift. If the addition or subtraction occurs after the function occurs, then it is a vertical shift. Absolute Value Function The absolute value function is defined by f(x) = |x|. If x ≥ 0 then the graph coincides with the line y = x. If x < 0 then the graph coincides with the line y = -x

y x f(x)=|x| Ex 1: Use the graph of y = |x| to sketch the graph of g(x) = |x – 1| - 2 y x 1 -2

Ex 2: Sketch the graph of h(x) = -|2x – 1| + 2 Note: Follow the order of operations to make the changes to your new graph. x y 4 things will happen to the graph of the parent function to get our new graph: The output of f(x) = |x| will be multiplied by a factor of 2 The graph will be shifted 1/2 unit to the right The graph will be shifted 2 units up. The graph will be reflected over the x-axis

Ex 3: Sketch the graph of f(x)= |2x – 6| + 1 Solution: Before you can see horizontal and vertical shifts, the leading coefficient must be 1. f(x)=| 2(x-3) | + 1 x y 3 1 y = |x| f(x)=|2x – 6| + 1

Ex: 4 Sketch the graph of Solution:We can see that we have a composition of functions (one function within another) Just like the order of operations we work from the inside out. Factor the quadratic to find the x-intercepts of the parabola. f(x) = | (x - 3)(x – 1) |x-int: (3, 0), (1, 0) Graph the parabola: you may need to complete the square to find the vertex. y x (connect the points to show the parabola)

Now, we take into account the second function in this problem…the absolute value function Recall that the absolute value measures the distance from zero, therefore, distance cannot be negative. Any output value that is negative will now become positive. y x 3 3 Now, connect your points with a smooth curve.

Square Root Function Parent function: The square root function is increasing on its entire domain. It has a minimum value of zero at x = 0. y x Connect your points with a smooth curve.

Ex 5: Sketch the graph of This graph is the graph of the parent function shifted two units to the right and one unit down. X-intercept: (3,0) Ex 6: Sketch the graph of The leading coefficient must be 1 before the horizontal shift can be seen. This graph is the graph from the last example reflected over the y-axis. Note: When the negative remains inside the function it is a reflection over the y-axis, if the negative is outside the function it is a reflection over the x-axis.

Greatest Integer Function Denoted: It is defined for a real number x to be the largest integer that is less than or equal to x. Ex: The greatest integer function has a wide application… Floor Function: used in computer science Denoted: Ceiling Function: Denoted: We round down We round up

The greatest integer function has a range with gaps and its graph “jumps.” The output of the greatest integer function is an integer… …and so on… The graph of the Greatest Integer Function y xGraph together

Again, the rules do not change for the shifting technique… Ex: Sketch the graph of Calculators (graphing) permit you to set the number of decimal places that you want your answer to round to. Such a function would look like… (computer science)