Pre-Calculus Chapter 1 Section 1 & 2 Modeling with Equations and Solving Functions and Their Properties 2013 - 2014.

Slides:

Advertisements

Similar presentations

SECTION 1.7 Graphs of Functions. T HE F UNDAMENTAL G RAPHING P RINCIPLE FOR F UNCTIONS The graph of a function f is the set of points which satisfy the.

Advertisements

1.2 Functions & their properties

Function Families Lesson 1-5.

College Algebra Chapter 2 Functions and Graphs.

Properties of Functions Section 1.6. Even functions f(-x) = f(x) Graph is symmetric with respect to the y-axis.

Symmetry Section 3.1. Symmetry Two Types of Symmetry:

Copyright © 2007 Pearson Education, Inc. Slide 2-1.

9/5/2006Pre-Calculus R R { [ 4,  ) } { (- , 3 ] } { R \ { 2 } } { R \ { 1 } } { R \ { -3, 0 } } R { (- 3,  ) } { (- , 4 ] U [ 2,  ) } { (- , -1)

Copyright © 2011 Pearson, Inc. 1.2 Functions and Their Properties.

Copyright © 2011 Pearson Education, Inc. Slide Graphs of Basic Functions and Relations Continuity (Informal Definition) A function is continuous.

Functions Definition A function from a set S to a set T is a rule that assigns to each element of S a unique element of T. We write f : S → T. Let S =

Continuity and end behavior of functions

R R { [ -8, ) } R { [ 0, ) } { [ 4, ) } { [ 0, ) } { (- , 3 ] }

Domain & Range Domain (D): is all the x values Range (R): is all the y values Must write D and R in interval notation To find domain algebraically set.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Section 3.3 Properties of Functions.

Section 2.3 Properties of Functions. For an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

FUNCTIONSFUNCTIONS Symmetric about the y axis Symmetric about the origin.

Pre Calculus Functions and Graphs. Functions A function is a relation where each element of the domain is paired with exactly one element of the range.

Today in Pre-Calculus Review Chapter 1 Go over quiz Make ups due by: Friday, May 22.

Pre-AP Pre-Calculus Chapter 1, Section 5 Parametric Relations and Inverses

1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Start-Up Day 2 Sketch the graph of the following functions.

Functions. Quick Review What you’ll learn about Numeric Models Algebraic Models Graphic Models The Zero Factor Property Problem Solving Grapher.

Lesson 2.6 Read: Pages Page 152: #1-37 (EOO), 47, 49, 51.

Section 4.1 Polynomial Functions. A polynomial function is a function of the form a n, a n-1,…, a 1, a 0 are real numbers n is a nonnegative integer D:

RATIONAL FUNCTIONS A rational function is a function of the form:

3.1 Symmetry; Graphing Key Equations. Symmetry A graph is said to be symmetric with respect to the x-axis if for every point (x,y) on the graph, the point.

Today in Precalculus Go over homework Notes: Graphs of Polar Equations Homework.

Section 1.2 Graphs of Equations in Two Variables.

Test an Equation for Symmetry Graph Key Equations Section 1.2.

Homework Text p. 388, #8-22 evens and #19 8) domain: -2, -1, 0, 1; 8) domain: -2, -1, 0, 1; range: -9, 2, 4, 5 range: -9, 2, 4, 5 10) domain: -4, -3, 2,

Henley Task teaches horizontal transformations Protein Bar Toss Part 1 teaches factoring if a ≠ 1 Section 3.4 for a = 1 Section 3.5 for a ≠ 1 Protein Bar.

Modeling and Equation Solving Section 1.1.  Mathematical structure that approximates phenomena for the purpose of studying or predicting behavior  The.

Pre-AP Pre-Calculus Chapter 2, Section 6 Graphs of Rational Functions

PRE-AP PRE-CALCULUS CHAPTER 3, SECTION 3 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS

 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.

Domain/Range/ Function Worksheet Warm Up Functions.

WARM UP: Linear Equations multiple choice Learning Targets :

Today in Pre-Calculus Go over homework Notes: Symmetry –Need a calculator Homework.

Chapter 1.2 Functions. Function Application The value of one variable often depends on the values of another. The area of a circle depends on its radius.

Date: 1.2 Functions And Their Properties A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain.

Mid-Term Review PRE-AP PRE-CALCULUS A. 0 B.3 C. -7 D. -6.

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

FUNCTIONSFUNCTIONS Symmetric about the y axis Symmetric about the origin.

1. Use the graph to determine intervals where the function is increasing, decreasing, and constant.

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:

AIM: What is symmetry? What are even and odd functions? Do Now : Find the x and y intercepts 1)y = x² + 3x 2) x = y² - 4 (3x + 1)² HW #3 – page 9 (#11-17,

Today in Pre-Calculus Do not need a calculator Review Chapter 1 Go over quiz Make ups due before: Friday, May 27.

Solving Inequalities Using Multiplication and Division Chapter 4 Section 3.

Chapter 4 – Graphing Linear Equations and Functions Algebra I A - Meeting 23 Example # 1 Step 1 : Make a Input-Output Table: Step 2 : List & Plot the Ordered.

College Algebra Chapter 2 Functions and Graphs Section 2.7 Analyzing Graphs of Functions and Piecewise- Defined Functions.

CHAPTER 1, SECTION 2 Functions and Graphs. Increasing and Decreasing Functions increasing functions rise from left to right  any two points within this.

Advanced Algebra/Trig Chapter2 Notes Analysis of Graphs of Functios.

Properties of Functions

Functions and Their Properties

Modeling and Equation Solving

Warm-up (10 min.) I. Factor the following expressions completely over the real numbers. 3x3 – 15x2 + 18x x4 + x2 – 20 II. Solve algebraically and graphically.

Properties of Functions

Standard MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2 + bx + c and f(x) = a(x – h)2 + k. c. Investigate and explain characteristics.

Chapter 2: Functions, Equations, and Graphs

College Algebra Chapter 2 Functions and Graphs

Warm-up (10 min.) I. Factor the following expressions completely over the real numbers. 3x3 – 15x2 + 18x x4 + x2 – 20 II. Solve algebraically and graphically.

1.2 Functions and Their Properties

Date Library of Functions

Properties of Functions

More Properties of Functions

Properties of Functions

AP Calculus Chapter 1, Section 5

Functions and Relations

Properties of Functions

Presentation transcript:

Pre-Calculus Chapter 1 Section 1 & 2 Modeling with Equations and Solving Functions and Their Properties

A pizzeria sells a rectangular 20” by 22” pizza for the same amount as a large round pizza (24” diameter). If both pizzas are the same thickness, which option gives you the most pizza for the money

The engineers at an auto manufacturer pay students $0.08 per mile plus $25 per day to road test their new vehicles. How much did the auto manufacturer pay Sally to drive 440 miles in one day? John earned $93 test-driving a new car in one day. How far did he drive?

Things you should know about Functions  Domain:  Range:  Function:  Vertical Line Test: Input values, x, independent Output values, y, dependent Each domain value has 1 y value A graph is a function if a vertical line passes through it and only intercepts at 1 point

Find the domain of the functions

Continuity  Continuous  Removable discontinuity  Jump discontinuity  Infinite discontinuity

Increasing and Decreasing Functions

For each function, tell the intervals on which it is increasing and decreasing.

Local and Absolute Extrema  Local values are located on an interval. Absolute values are the highest or lowest on the whole graph  Local maximum is the highest point in a section of a graph. If it is actually the highest point, it is the absolute maximum.

Symmetric about the y-axis xy

Symmetric about the x-axis xy These are not true functions because they fail the vertical line test. You can say (x, -y) is on the graph when (x, y) is on the graph.

Symmetric about the origin xy

Checking symmetry  To check if a function is an even function, subsitute (-x) in for x. If the function is the same, it is even.  To check if a function is odd, substitute (-x) in for x. If the function is the opposite sign of the original function, it is odd.  If the rules applied does not fit an even nor odd function, you would say the function is neither.

Asymptotes  An asymptote is an imaginary line where the function does not exist. It can forever get closer to that line but will never actually touch the line.

Finding Asymptotes

Before you leave today:  Complete #79 from page 104

Homework  Ch 1.1; Pg : 1-10, 22, 29, 31  Ch 1.2; Pg : 1-25 every other odd, 41 – 61 every other odd, 73