Functions: Even/Odd/Neither

Slides:

Advertisements

Similar presentations

Checking an equation for symmetry

Advertisements

WARM UP Zeros: Domain: Range: Relative Maximum: Relative Minimum:

Unit 6 Lesson #1 Intercepts and Symmetry

Symmetry of Functions Even, Odd, or Neither?. Even Functions All exponents are even. May contain a constant. f(x) = f(-x) Symmetric about the y-axis.

Even and Odd Functions madelinemontavon.cmswiki.wikispaces.net September 30th.

FUNCTIONSFUNCTIONS Symmetric about the y axis Symmetric about the origin.

Objective: Identify even or odd functions. Warm up a.Describe where is the function increasing, decreasing or constant. b.What is the relative maximum?

3-1 Symmetry & Coordinate Graphs Objective: 1. To determine symmetry of a graph using algebraic tests. 2. To determine if a function is even or odd.

Chapter 1 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc More on Functions and Their Graphs.

Chapter 2 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc More on Functions and Their Graphs.

Equal distance from origin.

Section 2.4 Symmetry Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Section 2.4. X-axis: replace y with –y. Simplify. If you get an equation = to what you started with, the function is symmetric to the x-axis. Y-axis:

Warm-Up. FUNCTIONSFUNCTIONS Symmetric about the y axis Symmetric about the origin.

Definition: Even Function

CHAPTER 2: More on Functions

2.1Intercepts;Symmetry;Graphing Key Equations

2.2 Graphs of Equations.

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

More on Functions and Their Graphs

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

Or ODD EVEN Functions.

Symmetry Section 2.7 Notes (part 2).

Warm up Find f(-3). 4 Find g(0).

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

3.1 Symmetry and Coordinate Graphs

Intercepts, Symmetry, Even/Odd and intersections

Graphs of Equations In Two Variables; Intercepts; Symmetry

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

3.1 Lesson Only! Skill Check

Points of Intersection

Integration by U-substitution

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Warm up Find f(-3). 4 Find g(0).

A. Symmetry with Respect to the Origin

Section 2.4 Symmetry.

4.3 Symmetry Which graph is NOT a function?

Warm up Find f(-3). 4 Find g(0).

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Chapter 3 – The Nature of Graphs

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

1-2 Day Two Even/Odd Functions

Analyzing Graphs of Functions and Relations Unit 1 Lesson 2

Section 2.4 Symmetry Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Skills Check Find f(-3). 4 Find g(0).

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

Power Functions Investigating symmetry to determine if a power function is even, odd, or neither.

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Chapter 2 More on Functions.

2.4 Symmetry and Transformations

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

Do Now Find the following: (f + g)(-1) = (g ͦ f)(x) = (g - f)(2) = (f /g)(x) =

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Warm up Factor..

Graphing Key Equations

Warm up Find f(-3). 4 Find g(0).

REFLECTIONS AND SYMMETRY

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Classify functions as even or odd.

Properties of Functions

Odd and Even Functions MCC9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of.

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Even and Odd Functions The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then.

Presentation transcript:

Functions: Even/Odd/Neither Math I: Unit 4 (Lesson 4)

Graphically… A function is even… If the graph is symmetrical about the y-axis, then it’s even. **Fold hotdog!

Graphically… A function is odd… If the graph is symmetrical about the y-axis & x-axis (or symmetrical about the origin), then it’s odd. **Fold hotdog & hamburger!

Examples: Graphically Neither Even Odd

The equations are exactly the SAME…so EVEN function. Algebraically… A function is even if f(-x) = f(x) Example 1: f(x) = 2x2 + 5 If you substitute in -x and get the SAME function that you started with, then it’s even. f(-x)= f(x) Replace x with –x 2x2 + 5 Simplify The new equation 2(-x)2 + 5 2(-x)(-x) + 5 2x2 + 5 The equations are exactly the SAME…so EVEN function.

Algebraically… A function is odd if f(-x) = -f(x) If you substitute in -x and get the OPPOSITE function (all the signs change),then it’s odd. Example: f(x) = 4x3 + 2x f(-x) f(x) Replace x with –x 4x3 + 2x Simplify The new equation 4(-x)3 + 2(-x) 4(-x)(-x)(-x) + 2(-x) -4x3 – 2x EVERY sign changed…so OPPOSITES… ODD function

Neither… Graphically… If a function does not have y-axis symmetry OR origin symmetry…then it has NEITHER. Algebraically… If, after substituting –x in place of x, the equation is not EXACTLY the same OR complete OPPOSITES, then the function is NEITHER.

Examples: Algebraically f(x) = x4 + x2 f(x) = 1 + x3 f(x) = 2x3 + x f(-x) f(x) (-x)4 + (-x)2 x4 + x2 (-x)(-x)(-x)(-x) + (-x)(-x) f(-x) f(x) 1 + (-x)3 1 + x3 1 + (-x)(-x)(-x) 1 – x3 f(-x) f(x) 2(-x)3 + (-x) 2x3 + x 2(-x)(-x)(-x) + (-x) 2x3+ x -2x3 - x SAME – so EVEN Not same and Not all signs changed – so NEITHER OPPOSITES– so ODD