Polynomial functions
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The document discusses polynomial functions of the form f(x) = anxn + ... + a1x + a0. It provides information to identify properties of a polynomial function based on its graph, including whether the highest term's exponent n is odd or even, the sign of the leading coefficient an, the lowest possible value of n, and the number of real zeros. It also gives rules for the behavior of graphs based on the sign of an and whether n is odd or even. The document concludes by listing the information needed to graph polynomial functions and providing two examples of functions to graph.
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This document discusses graphs of polynomial functions. It defines a polynomial function as a function of the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where n is a nonnegative integer and each ai is a real number. The degree of the polynomial is n and the leading coefficient is an. Polynomial graphs are continuous and smooth without breaks or cusps. The behavior of graphs as x approaches positive or negative infinity depends on the sign of the leading coefficient and whether the degree is odd or even. Polynomials can have real zeros where they intersect the x-axis. Their graphs may have up to n turning points and n zeros.
Unit 2.6
The document discusses rational functions and their graphs. It defines rational functions as the ratio of two polynomial functions. Key aspects covered include:
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Polynomial functions
A polynomial function is a function that can be written as f(x)=axn+bxn-1+cxn-2...gx+h, where a, b, c, ..., g, and h are real numbers and n is a positive integer. The key properties of a polynomial function are its degree, leading coefficient, factors, zeros, and relationship between factors and x-intercepts. The graph of a polynomial cannot intersect the x-axis more times than the number of distinct factors it has.
Jan. 13 polynonial sketching
The document discusses how to sketch graphs of polynomial functions. It explains that for odd degree polynomials, the graph behavior is opposite on the left and right sides, while for even degree polynomials the behavior is similar on both sides. It provides examples of how the graphs of cubic, quartic and quintic polynomials will appear. It outlines steps for sketching a polynomial graph which include finding the y-intercept, roots, sign of the function over intervals, and then sketching the graph. An example problem is given to factor a polynomial and sketch its graph.
Functions (Theory)
The document defines and provides properties of various mathematical functions including:
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The document discusses factorable polynomials and how to graph them. It defines a factorable polynomial as one that can be written as the product of linear factors using real numbers. For large values of x, the leading term of a polynomial dominates so the graph resembles that of the leading term. To graph a factorable polynomial, one first graphs the individual factors like x^n and then combines them, which gives smooth curves tending to the graphs of the leading terms for large x.
13 graphs of factorable polynomials x
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
Higher Maths 2.1.1 - Polynomials
The document provides an overview of polynomials, including definitions, examples, and key concepts. Some key points covered include:
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exponential functions (copied)
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Polynomial functions
This document discusses polynomial functions, which can be written as the sum of terms consisting of a number multiplied by a power of x, where the highest power is the degree. Important terms are the degree, leading coefficient, and standard form. Polynomials have as many x-intercepts as zeros, and the number of intercepts cannot exceed the degree. Examples of polynomials and non-polynomials are provided.
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This document discusses exponential functions. It begins by stating the objectives of understanding properties of exponents, simplifying exponential expressions, solving exponential equations, and sketching graphs of exponential functions. It then defines exponential functions and discusses their general form. Several properties of exponents are listed and examples of simplifying expressions using these properties are shown. The document also demonstrates how to solve various exponential equations. Finally, it discusses graphing and properties of exponential functions, including the special number e and the exponential function with base e.
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orca_share_media1680312384648_7047740956181238360.pptx
This document discusses piecewise functions and even and odd functions. It defines a piecewise function as a function with multiple sub-functions where each applies to a certain interval of the domain. It provides an example of finding values of a piecewise function at specific x values. It also outlines the steps to graph a piecewise function. The document then defines even functions as those where f(x) = f(-x) and odd functions as those where -f(x) = f(-x). It provides examples of even and odd functions and notes that most functions are neither. It concludes with some properties of even and odd functions.
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The document discusses techniques for finding zeros of polynomial functions, including the rational zero theorem, synthetic division, and Descartes' rule of signs. It provides examples of using these methods to find the rational zeros of various polynomial functions and determining the total number of positive and negative zeros based on the signs of the coefficients.
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This document discusses polynomial functions. It defines polynomial functions as functions of the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where the exponents are whole numbers. It describes the leading coefficient, constant term, and degree of a polynomial function. Common types of polynomial functions like constant, linear, quadratic, cubic, and quartic functions are summarized. Methods for identifying if a function is a polynomial function, evaluating polynomial functions using direct substitution and synthetic substitution, and determining the left and right behavior of a polynomial function's graph are also discussed.
6.3 evaluating-and-graphing-polynomila-functions
This document discusses polynomial functions. It defines polynomial functions as functions of the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where the exponents are whole numbers. It describes the leading coefficient, constant term, and degree of a polynomial function. Common types of polynomial functions like constant, linear, quadratic, cubic, and quartic functions are identified based on their degree. Methods for identifying if a function is a polynomial function, evaluating polynomial functions using direct substitution and synthetic substitution, and determining the left and right behavior of a polynomial function's graph are also discussed.
Functions
The document defines functions, relations, domains, ranges, and different types of functions such as even, odd, and composite functions. It provides examples of evaluating functions at given values, performing operations on functions, and composing functions. Graphs of functions and their properties such as the vertical line test are also discussed. Homework problems involve identifying functions, finding their domains and ranges, evaluating, operating on, and composing various functions.
G10_Daily Lesson Log_Second QUARTER.docx
The document outlines the daily lesson log for a Grade 10 mathematics class. It includes 4 sessions on the topic of polynomial functions. The objectives are to illustrate, find intercepts of, and graph polynomial functions. Examples are provided for identifying key features of polynomials like degree, leading coefficient, constant term, and finding x- and y-intercepts by setting the polynomial equal to 0 or using the factored form. The behavior of graphs is also discussed based on whether the degree is odd or even, and the sign of the leading coefficient.
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This document discusses the axioms that define a field and the set of real numbers. It outlines the closure, associativity, commutativity, distributive property, identity element, and inverse element axioms for fields. It also covers the equality axioms of reflexivity, symmetry, transitivity, and property extensions for addition and multiplication. The goal is to define the necessary axioms for the set of real numbers to form a field.
Math 4 axioms on the set of real numbers
The document discusses the axioms that define a field and the set of real numbers. It outlines the closure, associativity, commutativity, distributive property, identity element, and inverse element axioms for fields. It also covers equality axioms and proves theorems like cancellation for addition and the involution property using the field axioms.
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The document defines upper and lower bounds of a set as numbers greater than or equal to and less than or equal to elements of the set, respectively. It also defines the least upper bound and greatest lower bound as the smallest upper bound and largest lower bound. Several examples of sets are given with their greatest lower and least upper bounds identified. The homework assigns finding the greatest lower and least upper bounds of additional sets.
Specific function examples
The document discusses different types of functions including linear, quadratic, absolute value, and square root functions. It provides the definitions and key properties of each function such as domain, range, intercepts, vertex, and transformations that modify the graph. Examples are worked through demonstrating how to find specific characteristics of each function and graph transformations.
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The document contains several passages that discuss the purpose and value of mathematics education. Some key points made are:
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This document contains 13 examples of permutations and combinations problems from a Mathematics 4 class. It provides example problems involving passcode combinations, palindromes, paths on a grid, lottery number selection, triangles in an octagon, quiz answers, multiple choice exams, hamburger toppings, NBA championship series outcomes, poker hands, and donut flavors. The document is intended to illustrate different applications of permutations and combinations in mathematics.
Permutations
1. The document contains examples and formulas for permutations and combinations. It discusses linear permutations, distinguishable permutations, circular permutations, and ring permutations.
2. One example asks how many five-letter words can be formed from a set of vowels and consonants if they must alternate with no repetition.
3. The combination formula is presented, and examples ask how to select teams and committees from groups of people.
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The document provides 12 examples of counting problems using the fundamental counting principle. It explains that if one event can occur in n ways and a second in m ways, the total number of outcomes is n*m. The examples include counting combinations of digits, tournament placements, outfit combinations, license plates, dice rolls, classroom seating arrangements, true/false tests, multiple choice tests, and card shuffles.
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The document provides information about different types of infinite sequences, including explicitly defined sequences, recursively defined sequences, arithmetic sequences, harmonic sequences, and geometric sequences. It gives examples and formulas for finding terms, sums, and analyzing properties of each type of sequence. Key points covered include the definitions of explicit and recursive sequences, the formulas for the nth term and partial sum of arithmetic and geometric sequences, and examples of finding terms and analyzing sequences.
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The document discusses operations on complex numbers, including multiplication, raising to powers, and De Moivre's theorem. It provides examples of multiplying complex numbers algebraically and in polar form. It also gives examples of raising complex numbers to powers using the rules zn = rnθn and De Moivre's theorem, which states that (r cis θ)n = rn cis nθ.
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The document discusses inverse trigonometric functions. It defines the inverse sine function as sin^-1x = arcsin(x), with domain [-1,1] and range [-π/2, π/2]. It provides examples of evaluating inverse trig functions like sin^-1(1/2) = π/6. The inverse cosine function is similarly defined as cos^-1x = arccos(x), with domain [-1,1] and range [0,π]. The document concludes with a short quiz evaluating inverse trig expressions.
Graphing trigonometric functions
The document discusses how to graph trigonometric functions like sine and cosine. It explains that the sine function has a range from -1 to 1 and is decreasing in quadrants 1 and 2. The cosine function has the same range and properties as sine, except its zeros occur at odd integer multiples of π/2. Both functions have an amplitude of 1 and a period of 2π. The document also describes how changing the coefficients in trigonometric functions affects their amplitude, period, and horizontal or vertical shifts.
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Circles - Degenerate and Null cases
This document discusses special cases of circle equations, including degenerate circles which represent a single point, and null sets which represent no points. It provides examples of converting circle equations to standard form, identifying degenerate and null sets, and finding the radius and equation of a circle given two points. Practice problems are also presented for students to identify circle equations as representing circles, points, or null sets, and to find the equation of a circle given its radius endpoints.
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This document provides examples and explanations for analyzing problems involving circles. It begins by working through an example of finding the equation of a circle tangent to a given line with a specified center. It then discusses using an algebraic approach to find the standard equation of a circle passing through three given points by setting up three equations and solving simultaneously for the center and radius. Finally, it covers using a geometric approach involving finding the perpendicular bisectors of chord midpoints and their intersection to determine the circle's center. The document aims to demonstrate different methods for solving circle geometry problems.
Circles Lecture - Part 1
The document provides examples and explanations of completing the square to derive the standard form of a circle equation. It defines a circle as all points equidistant from a center point, and discusses radius, diameter, and graphing circles. Examples demonstrate finding the equation, center, and radius of circles given algebraic or geometric information. The general form of a circle equation is derived.
Math 1 Orientation Presentation
Math 1 renews arithmetic concepts and skills through problem solving, starting with sets and leading to evaluating and operating on algebraic expressions, including solving word problems involving linear and rational expressions. The course covers sets, real number systems, algebraic expressions, linear equations, word problems, linear inequalities, absolute values, polynomials, rational expressions, radicals, and is graded based on daily participation, homework, quizzes, projects, exams, and a periodic exam.
Math 1 Course Outline 2009-2010
This document outlines the course content and grading components for a Math 1 course. The course covers topics like sets, real number systems, algebraic expressions, linear equations, inequalities, polynomials, rational expressions, and radicals. Students' grades are based on class standing (70%), recitation (5%), homework (15%), quizzes (20%), long tests (30%), and a periodic exam (30%). There are three long tests and a final exam each quarter to evaluate students' understanding of the course material.
Math 3 Student Orientation Presentation
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Polynomial functions
- 1. Polynomial Functions PSHS Main Campus July 9, 2012 PSHS Main Campus () Polynomial Functions July 9, 2012 1/4
- 2. Polynomial Functions f (x) = an xn + ... + a1 x + a0 Given the graph on the right, identify: 1 If n is odd or even. 2 The sign of an . 3 The lowest possible value of n. 4 The number of real zeroes of the function. PSHS Main Campus () Polynomial Functions July 9, 2012 2/4
- 3. Polynomial Functions f (x) = an xn + ... + a1 x + a0 Given the graph on the right, identify: 1 If n is odd or even. 2 The sign of an . 3 The lowest possible value of n. 4 The number of real zeroes of the function. PSHS Main Campus () Polynomial Functions July 9, 2012 2/4
- 4. Polynomial Functions f (x) = an xn + ... + a1 x + a0 Given the graph on the right, identify: 1 If n is odd or even. 2 The sign of an . 3 The lowest possible value of n. 4 The number of real zeroes of the function. PSHS Main Campus () Polynomial Functions July 9, 2012 2/4
- 5. Polynomial Functions f (x) = an xn + ... + a1 x + a0 Given the graph on the right, identify: 1 If n is odd or even. 2 The sign of an . 3 The lowest possible value of n. 4 The number of real zeroes of the function. PSHS Main Campus () Polynomial Functions July 9, 2012 2/4
- 6. Polynomial Functions f (x) = an xn + ... + a1 x + a0 Given the graph on the right, identify: 1 If n is odd or even. 2 The sign of an . 3 The lowest possible value of n. 4 The number of real zeroes of the function. PSHS Main Campus () Polynomial Functions July 9, 2012 2/4
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