Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers less than 6 Solution: the order in which the elements are listed is not important So the previous set can be written as. { } or any other order 3 is an element of the previous set so 3 { } and we read it 6 does not belong to the set { so 6 { } A Set is named using a capital Letter Ways of writing sets 1. by writing all elements ex: A ={ } by describing the elements (set notation) A={ } Definition: 1. Finite set : is a set that has limited number of elements. ex: { } Section 22 Page 1
Infinite set: is a set that has an unending list of distinct elements Ex: {xlxis * Natural numbers or counting numbers sets are often written using a variable {x x is a natural number between 1 and 7} when discussing a particular problem we can usually identify a universal set that contains all the elements appearing in any set in the given problem also we have the null set (or empty set) containing no element, it is written as { } or but not Definition: A set A is a subset of set B if all elements of A are in B and we write it as and read A is a subset of B ex: Remark: If A is any set then Ex: Is it true * Equal sets: Two sets are equal if they have the same elements A=B if and only if and Section 22 Page 2
A=B if and only if and ex: Ex: If and the universal set represent these sets in a Venn diagram * operations on sets - 1. The Complement of a set Given a set A and a universal set U then the complement of A is denoted by A' Shade. A U Ex: Let 1. Find 3. 4. The intersection of sets : are the elements belonging to both sets A intersection B is denoted by Section 22 Page 3
3. The union of sets: are all the elements that belongs to A or B without repetition A B Ex: Find I. AUB Definition: Disjoint sets A and B are disjoint sets iff Ex: Ex: True or false 1. 3. 4. 5. 6. 7. 8. Ex: let U be the universal set where U={ all whole numbers less than 11} Section 22 Page 4
A = { all even natural numbers less than or equal to 8} B={2,4,5,8,10} Find the elements of 1.U= 3. 4. 5. A= A' = A B = AUB'= Ex: If A = {1, 3, 5, 7, 9), B={0,2,4,6,10} c={1,4,9,16,25} Find Ex: which one of the following statements is false for the sets A,B and C given the adjacent figure A :! 3. C 4. 5. Ex: List all elements of the following sets 1. Section 22 Page 5
Ex: List all elements of the following sets 1. Ex: Given Find Section 22 Page 6
R2: Real Numbers and their Properties * sets of numbers 1. Natural numbers: N= {1, 2,3,4,5,... } Whole numbers = { 0,1, 2, 3, 4,...} 3. Integers: =Z 4. Rational numbers Q Is the set-of numbers that can be written in the form where 1. a,b are integers. Types of rational numbers 1 Natural numbers since whole numbers o= 3.Integers -2= 4. common fractions: 5. terminating decimals : 6. repeating decimals 7. roots that can be simplified to one of the above types 5. Irrational numbers: Q' Types of irrational numbers 1. Non repeating nor terminating decimals Section 22 Page 7
roots that cannot be simplified to a number without the root sign in it 3. Special numbers like 6. Real numbers: R=QUQ' Ex: let List the elements from set A that belong to each set 1 Natural numbers Whole numbers 3. integers : 4. rational numbers 5. irrational numbers. 6. real numbers Ex: Decide whether each statement is true or false 1. Every integer is a whole number Every natural number is an integer 3.Every irrational number is an integer Section 22 Page 8
4.Every integer is rational 5.some rational numbers are irrational 6.Some real numbers are integers 7.The sum of two rational numbers is irrational 8.The sum of any two irrational numbers is irrational Ex: Given A= how many rational numbers are in A? solution: * Exponents: If n is any positive integer and a is any real number then the nth power of a is power (exponent) n times base Ex: Evaluate: 3. 4. Order of operations: steps of evaluating numeric expressions 1. work separately above and below each fraction work brackets from inner to outer 3. work powers and roots 4.work multiplication and division from left to right 5.work addition and subtraction from left to right Section 22 Page 9
Ex: Evaluate: 1. Ex: Evaluate each expression if I. 2 * properties of real numbers for any real numbers 1. Closure property a,b and c Commutative property is a real number Is a real number 3. Associative property 4. Identity property o is the identity of addition element 1 is the identity element of multiplication Section 22 Page 10
5. Inverse property of multiplication a,-a are additive inverses 6. Distributive property where are multiplicative inverses Ex: Evaluate: * Absolute value for any real number a Ex: Evaluate 1. 3. if *properties: for all real numbers a and b 1. 3. 4. Section 22 Page 11
3. 4. 5. (triangle inequality) Ex: let m=13, n= -9 Evaluate Ex: Redefine 1 - Ex: If Simplify Ex: If simplify Section 22 Page 12
* Distance between two points on the number line If P and a are two points on the number line with coordinates respectively then the distance between them is a and b Ex: Find the distance between -5 and 8 Ex: Is there an associative property for subtraction Solution: Ex: If / / find Ex: True or False I. Every natural number is either prime or composite Section 22 Page 13
I. Every natural number is either prime or composite 3. For every real number X; 4. 5. Every nonzero real number has a multiplicative inverse Ex: If simplify Ex: Evaluate Ex: If simplify Ex: Simplify Section 22 Page 14
Ex: Simplify Ex: If then write the expression without the absolute value symbols Section 22 Page 15
Ex: true or False 1. The product of two prime numbers is a composite number every rational number has a multiplicative inverse 3-The sum of two prime numbers is prime 4. The product of two irrational numbers is irrational Section 22 Page 16
R3 Polynomials *Rules of exponents I. 3 4.. 5. 6. where 7. Ex: Simplify assume all variables represent nonzero real numbers = An algebraic term is a number or a product of a number with a variable or more than one variable ex: term not a term Section 22 Page 17
the constant number in the term is called Coefficient. * Like terms: Two terms are alike if they contain the same variables with the same powers ex: Like terms unlike terms Note! We can add or subtract like terms only = = * Algebraic expression: is an algebraic term or a sum, product, division of algebraic terms Ex: polynomial: is a sum of a finite number of terms such that ex: polynomial I. the powers of the variables are nonnegative integers the Coefficients of the variables are real number, not a polynomial Degree of a polynomial Section 22 Page 18
Degree of a term 1. In one variable the degree is the exponent of the variable ex Term degree In more than one variable the degree is the sum of the powers of the variables ex: Term degree. Degree of a polynomial is the greatest degree of the terms in a polynomial Ex: Find the degree of each polynomial Polynomial Degree Definition 1. Monomial is a polynomial of one term only Binomial is a polynomial with two terms only Section 22 Page 19
3. trinomial is a polynomial with 3 terms only * Addition and subtraction of polynomials we can add or subtract coefficients of like terms only Ex: Add or subtract Multiplication: Ex: Multiply 1. Special products 1 Ex: find the product 1. 3. Section 22 Page 20
Note: 4. 5. 1. special products 2 Ex: Simplify: Ex: find the coefficient of x in the polynomial Ex find the coefficient of in the polynomial is Section 22 Page 21
Ex: Simplify: Ex: Simplify: Ex: find the sum of the coefficients of and in the expression Ex: find the degree of the polynomial Ex: Simplify Ex: find the product * Division: dividing polynomials Ex: Divide 65 7 Section 22 Page 22
Ex: Divide by Ex: Divide by Ex: If then find the values of m and n Section 22 Page 23
Section 22 Page 24
R4 Factoring Polynomials Definition: Factoring a polynomial means writing the polynomial as a product of polynomials each of degree original degree Methods of factoring 1. Factoring the greatest common factor Ex: Factor = Factoring by grouping Ex: Factor = Section 22 Page 25
3. Factoring Trinomials a. If coefficient of Ex: Factor: = = = b. If coefficient of Ex: Factor: = = = = = Section 22 Page 26
4. Factoring a perfect square Ex: Factor 5. Factoring Binomials I. Difference between two squares Ex: Factor Section 22 Page 27
= Difference or sum of two cubes Ex: Factor = *factoring by substitution Ex: factor = 3. Section 22 Page 28
Ex: Factor Ex: Find the values of b that makes a perfect square Section 22 Page 29
Ex: Find the values of c that makes a perfect square Ex: factor Ex: find the value of K that makes the trinomial a perfect square Ex: factor Section 22 Page 30
Section 22 Page 31 R5 Rational Expressions Definition: rational expression is the quotient of two polynomials P and Q with Domain of rational expressions _ = {zeros of the denominator} Ex: find the domain of: 1. Domain = Domain = * Lowest terms of rational expressions a rational expression is writer in Lowest terms if the greatest Common factor of its numerator and denominator = 1 Rule: where Ex write each rational expression in lowest terms I. 3. 4. 5. 6.
Section 22 Page 32 6. Multiplication and division where where Ex: Multiply or divide 1. 3. 4. 5. 6. Addition and subtraction Note: Finding the Least Common denominator(lcd) Step1: Write each denominator as a product of prime factors Step2: form a product of all different prime factors, each factor should have an exponent as the greatest exponent that appear in the factor
Section 22 Page 33 Ex: Add or subtract 1. 3. 4. * Complex Fractions: is the quotient of two rational expression Ex: Simplify each Complex fraction: I. 3.
Section 22 Page 34 3. 4. 5. 6. Ex: Simplify Ex: simplify
Section 22 Page 35 Ex: Simplify Ex: find the LCD of Ex: Simplify Ex: Simplify
Section 22 Page 36 Ex: Simplify Ex: Simplify Ex: simplify Ex: Simplify
Ex Simplify: Section 22 Page 37
Section 22 Page 38 R6: Rational Exponents * Negative exponents If and then Ex: Evaluate without using negative powers, assume all variables represent nonzero real numbers I. 3. 4. 5. * Quotient Rule: for all integers m and n and all nonzero real number Ex: Simplify each expression, assume all variables represent nonzero real numbers 1. 3. Ex: Simplify each expression without using negative exponents, assume all variables represent nonzero real numbers 1. 3.
* Rational exponents 1. n is even If n is an even positive integer and if then is the positive real number whose nth power is a i.e. If n is an even positive integer and is not a real number (Complex number) then 3. if n is an odd positive integer and a is any real number then is areal number Ex: Evaluate each expression I. 3. 4. 5. 6. 7. * Rational exponents for all integers m, all positive integers n and all real numbers a for which is areal number Section 22 Page 39
Section 22 Page 40 Ex: Evaluate: 1. Ex: simplify each expression, assume all variables represent positive real numbers 1. 3. 4. Ex: factor out the Least power of the variable 1. 3.
Section 22 Page 41 Ex: Simplify 1. 2 Ex: Simplify: 1.
Section 22 Page 42 Ex: Factor: 1. Ex: Calculate mentally 1. Ex: Simplify if
Section 22 Page 43 Ex: Simplify: Ex, Simplify = Ex: If simplify Ex: simplify where : Ex: Simplify
Section 22 Page 44 Ex: simplify Ex: Simplify Ex: Simplify
Section 22 Page 45 R7: Radical Expressions *Radical notation If a is a real number, n is appositive integer and index is a real number then radical sign radicand Ex: Write each using exponents 1. Ex: Write in radical form 1 Ex: Write in exponential form 1. Note: Rule: 1. If n is an even positive integer then If n is an odd positive integer then Ex: Simplify: 1.
Section 22 Page 46 1. 3. 4. 5. 6. 7. Rules of radicals 1. 3 Ex: Simplify: 1. 3. 4. 5.
Section 22 Page 47 5. Simplified radicals : the rules are in the book page 66 Ex: Simplify: 1. 3. * Operations with radicals Like radicals: are radicals with the same radicand and same index Like radicals unlike radicals Note: Only Like radicals can be added or subtracted Ex: Add or subtract, assume all variables represent positive real numbers 1. 3.
Section 22 Page 48 3. Ex: Simplify: 1. 3. Ex: find each product 1. 3. Rationalizing the denominator (denominator with one term) 1. r 3.
Section 22 Page 49 4. 5. Ex: Simplify, assume all variables are positive real numbers 1. Ex: Rationalize the denominator ( denominator with two terms) + 1. 3. 4. 5.
Section 22 Page 50 (denominator with 3 terms ) 6. Ex: simplify 1. 3. 4. 5. 6. 7. 8. Ex: Simplify: 1.
Section 22 Page 51 3. 4. 5. 6. 7. 8. 9.
Section 22 Page 52 Ex: If and find M+N Ex: Simplify: Ex: simplify: Ex: Simplify: 1. Ex. Simplify
Section 22 Page 53 1.3 Complex Numbers is called the imaginary unit Numbers of the form are called complex numbers where : real part : complex part Note: equality of complex numbers if and only if and In the complex number If which is areal number so the set of real numbers is a subset st the complex numbers If is said to be a pure imaginary number Standard form of a complex number is If then Ex: Write as a product of a real number and 1. 3. Operations on complex numbers Note, when dealing with negative radicands, use the definition
Section 22 Page 54 before using any other rules of radicals Ex: but so Ex: Multiply or divide as indicated simplify each answer Ex: Write in standard form Addition and subtraction of complex numbers Ex: find the sum or difference 2
Section 22 Page 55 Multiplication of complex numbers Ex: find the product 1. 3. * powers of So Ex: find
Section 22 Page 56 The product of a complex number with its conjugate Ex: Write each quotient in standard form I. Ex: Simplify 1. 3. 4. 5. 6. 7. 8.
Section 22 Page 57 Ex: Evaluate if Ex: find the conjugate of the complex number Ex: find the imaginary port of the complex number Ex: If where find
Section 22 Page 58 Ex: If find Ex: If A is the real part and B is the imaginary part of the complex number find Ex: find the conjugate of the complex number Ex: find the sum of the real part and the imaginary part of the Complex number where
Section 22 Page 59 Ex: find the sum of the real part and the imaginary part of the Complex number where Ex: find the conjugate of Ex: Simplify
Section 22 Page 60 Ex: find the conjugate of Ex: find the real and imaginary parts of the complex number Ex: Write the complex number in standard form Ex: Simplify
Section 22 Page 61 1.1 Equations and Inequalities Definition of an equation: is a statement that two expressions are equal ex:, Solving an equation means to find all numbers that makes the equation a true statement are called solutions or roots) (these numbers Equivalent equations: Are equations with the same solution set * Addition and multiplication properties of equally If 1. If then If and then * Solving linear equations Definition: a linear equation in one variable Is an equation that can be written in the form i.e. one variable with power =1 Ex: Solve 1.
Section 22 Page 62 * Types of equations 1. An identity : is an equation which is always true i.e.: true for any real number ex: A contradiction: is an Equation which is satisfied by some numbers but not by others ex: true if but false if 3. A contradiction an equation that has no solution ex: Note: If solving an equation Leads to 1. 0=0 or number = it self it is an identity 3. x = number number = different number it is a conditional its a contradiction
Section 22 Page 63 Ex: decide whether each equation is an identity s conditional or a contradiction, give the solution set if any 1, 3. * Solving for a specific variable (literal equations) means making the specified variable alone on one side of the equation and every thing else on the other side Ex: Solve for the specified variable I., Solve for t, solve for K
Section 22 Page 64 3. Solve for y 4.,solve for x Ex:If is a solution of the equation find k Ex: If, solve for P Ex: Solve
Section 22 Page 65 Ex: If EX: Which of the following statements is True 1. 3. and is a contradiction is an identity are equivalent equations 4. is a conditioned equation 5. is a solution of the equation Ex: If 'solve for z Ex: If the equation find
Section 22 Page 66 Ex: If the equation is an identity find Ex: Solve for z Ex: If solve for B
Section 22 Page 67 1.2 Applications and Modeling with linear equations Ex: The length of a rectangle is 2 inches more than the width. If the length and width are increased by 3 inches, the perimeter of the new rectangle will be 4 inches less than 8 times the width of the original rectangle. Find the dimensions of the original rectangle Ex: A puzzle piece in the shape of a triangle has perimeter 30 Cm. Two sides of the triangle are each twice as long as the shortest side. Find the length of the shortest side Ex: The perimeter of a rectangle is 22 Cm. If the length of the rectangle is 1 cm less than twice the width then find the length
Section 22 Page 68 Ex: If the difference between 7 times a fraction (in simplest form) and 1 is. find the sum of the numerator and denominator Ex: The perimeter of a rectangular piece of land is 3400 feet. The length of the land is 200 feet more than twice the width. Find the area of the Land in square feet.
Section 22 Page 69 Ex: In the adjacent figure, the area of the big rectangle is 24 find the area of the shaded region Ex: A triangle has a perimeter of 77 Cm, each of the two Longer sides is 3 times as long as the shortest
side of the triangle. Find the length of each Side Section 22 Page 70
Section 22 Page 71 1.4 Quadratic Equations A Quadratic equation in one variable can be written in the form and this form is called the standard form where or its a second degree equation ie: an equation with a squared variable term and no terms of greater degree Quadratic equations Non quadratic equation, * Solving a quadratic equation 1. by factoring and 1. Ex: solve 2 3. Square root property If then or
Section 22 Page 72 Ex: Solve 1. 3. 3. Completing the squares To solve by completing squares 1. If divide both sides by a make the constant term alone on one side 3.square half of the coefficient of x and add this square to both sides of the equation 4.factor the resulting trinomial as a perfect square 5. use the square root property Ex: Solve by completing the squares 1.
Section 22 Page 73 4. by the quadrate formula The solution of the quadratic equation where are Ex: Solve by the quadratic formula I.
Section 22 Page 74 * Solving a Cubic equation Ex: Solve I. * Solving for a specified variable Ex: Solve for the specified variable I. solve for solve for y
Section 22 Page 75 * The discriminant discriminant = the value of the discriminant can be used to determine whether the solutions of a quadratic equation are rational, irrational, or non real complex numbers So If are integers then Discriminant Number of solutions Type of solutions positive perfect square Two Rational positive not perfect square Two Irrational zero One (double zero) Rational Negative Two Non real complex real Ex: Determine the number of distinct solutions sand tell whether they are rational irrational or non real complex I. 3.
Section 22 Page 76 3. Ex: Solve a) for x in terms of y b) for y in terms of x
Section 22 Page 77 Ex: find a quadratic equation that has the given solutions 1. 3. Ex: If the equation is written in the completing square form find Ex: Find the number and type of solution, of the equation
Ex: If the and quadratic equation has sum of Solutions and product of solutions find Ex: Find the sum of the solutions of the equation Ex: Determine the number and type of solutions of the equation Section 22 Page 78
Section 22 Page 79 Ex: Find the sum of the values of for which the equation has only one solution
Section 22 Page 80 1.6: Other types of Equations and Applications * Rational equations: is an equation that has a rational expression for one or more term, Note: zeros of the denominator must be excluded from the Solution set Ex: solve I. 3. 4.
Section 22 Page 81 Equations with radicals Power property: If p and a are algebraic expressions thon every solution of the equation is a solution of the equation for any positive integer n Note: It is very important to check the solutions in the equation after Solving Ex: Solve 1. 3. Note: No need to check if all indexes of the radical are odd
Section 22 Page 82 * Equations quadratic in form Ex: Solve I. 3. 4.
Section 22 Page 83 Ex: Solve I. 3.
Section 22 Page 84 4. 5. Q: Solve the equation Q: Solve Q: If and are the real Solutions of the equation
Section 22 Page 85 then Find Q: Solve Q: Find the sum of the real solutions of
Section 22 Page 86 1.7 Inequalities Definition: Inequality is a comparison between two quantities using,,, A value of a variable for which the inequality is true is a solution Two inequalities are equivalent it they have the same solution set Properties of inequalities For real number a,b and c I. If then If and then 3. If and then Note: always remember to reverse the direction of the inequality symbol when multiplying or dividing by a negative number * Linear inequalities in one variable is an inequality that can be written in the form where any of the symbols, Types of intervals may be used
Section 22 Page 87 Ex: solve give the solution set in interval notation 1. 3.
Section 22 Page 88 Quadratic inequalities A quadratic inequality is an inequality that can be written in the form where can be replaced by * Solving quadrant inequalities: Ex: solve Ex Solve
Section 22 Page 89 Rational inequalities: Ex: solve Ex: solve
Section 22 Page 90 Ex: Solve Ex: Solve Ex: Solve Ex, Solve
Section 22 Page 91 Ex: Solve Ex: solve Ex: Solve
Exe Solve Section 22 Page 92
Section 22 Page 93 Ex: Solve Ex: solve
Section 22 Page 94 1.8 Absolute value Equations and inequalities The absolute value of a written gives the distance from (a) to (zero) So means or Properties of absolute value 1. For iff or 3. iff For any positive number b iff or 4. iff or * Absolute value equations: Ex. Solve each equation 1.
Section 22 Page 95 * Absolute value inequalities Ex: Solve 1. 3. Ex: Solve each inequality: I.
Section 22 Page 96 3. Ex: Write each statement using an algebraic value inequality 1. m is no more than 9 units from t is within 0-02 units of 5.8 Ex: Suppose that and we want y to be within 0-001 units of 6. For what values of x will this be true?
Section 22 Page 97 Ex: Solve: 1. 3. 4.
Section 22 Page 98 5. 6. 7. 8. 9. 10. 11. 1
Section 22 Page 99 1 13. Ex: If A is the Solution set of and B is the solutions of then find Ex: Solve
Section 22 Page 100 Ex: Find the sum of the solutions of the equation Ex: If is equivalent to find m and n Ex.. Find the so hit, on set of the compound inequality and
Ex: Solve Section 22 Page 101
Section 22 Page 102 2-1 Rectangular coordinates and Graphs * ordered pair: is a pairing of one quantity with another * The rectangular coordinate system (Cartesian coordinate system) or xy-plane or coordinate plane 2nd quadrant 1st quadrant origin 3rd quadrant 4th quadrant Distance formula: If by then the distance between P and R written is given Ex: Find the distance between Ex: Are the points and represent vertices of a right triangle Ex: Are the points Solution, and collinear?
Section 22 Page 103 Solution, and collinear Collinear means that they are on the same line *The midpoint formula. The midpoint of the line segment with endpoints and has coordinates Ex: Use the midpoint formula to do each of the following a) Find the coordinates of the midpoint end points M of the line segment with b) Find the coordinates of the other endpoint B of a segment with one end point and a midpoint Ex: For each equation, find at least three ordered pairs that are solutions a) b)
Section 22 Page 104 c) * Graphing an equation by point plotting Ex: Draw each equation: a) b) c)
Section 22 Page 105 EX: Find the coordinates of the other endpoint of the segment given its midpoint and one end point is Ex. If the point is in the second quadrant in what quadrant is 1) 2) 3) 4) Ex: Show that the points are the vertices of a rhombus Ex: If the distance between the points and is 5, find all possible values of x
Section 22 Page 106 Ex: If is the midpoint of the line segment joining and find Ex: If find the distance between and Ex: Find the coordinates of all points on the y-axis that are 5 units from the point Ex: In the adjacent figure if then find the height of the triangle
Section 22 Page 107
Section 22 Page 108 2 Circles A circle is the set of all points in a plane that have the same distance ( called radius) from a fixed point (called the center) a circle with center and radius has equation radius Center which is the center radius form Ex: Find the center-radius form of the equation of each circle I. center radius= Center radius=2 Ex: Graph each circle I. * General form of the equation of a circle for some real numbers c,d and e can have
Section 22 Page 109 a graph that is a circle or a point or is non existent if we complete the squares of this equation we get for some number m I. If If 3.If then the graph is a circle then the graph is a point then the graph is nonexistent Ex: Determine the type of each graph 1. 3. 4. 5.
Section 22 Page 110 6. 7. Ex: Find the equation of the circle of center and tangent to the x-axis Ex: Find the equation of the circle with center point 'passing through the
Section 22 Page 111
Section 22 Page 112
Section 22 Page 113
3 Functions: In an ordered pair x : is the independent variable : is the dependent variable Definition: A relation is a set of ordered pairs x-coordinate first component y -coordinate second component Definition: A Function is a relation in which for each x there is only one y related to it Ex: Decide whether each relation defines a funetim 1. 3. Definition: Domain is the set of all first components of the ordered pair Range is the set of all second components of the ordered pair Ex. Give the domain and range of each relation and tell whether its a function or not 1. Domain = Range = Domain = Range = 3. Domain= Range = Section 22 Page 114
4. Domain= range = 5. Domain= Range = 6. Domain = Range = 7. Section 22 Page 115
Domain = Range = Note: Unless specified otherwise, the domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable Excluded values from the domain I. zeros of the denominator values that make complex numbers Vertical Cine test to determine functions from graph If each vertical line intersects a graph in at most one point, then the graph is a function Ex: Use the vertical line test to determine whether each graph in the previous example is a function or not Ex: Decide whether each relation defines a function and give the domain and range 1. ' Section 22 Page 116
3. 4. 5. 6. 7. Section 22 Page 117
Note: In most of the relations the ones that will not be functions are: 1. If y is raised to an even power If we have in the formula 3. If y is in an inequality * Function notation If a relation is a function we can replace by the symbols. so read (f) of (x) it doesn't mean f times x Ex: Let Find and simplify each of the following I. 3. Ex: For each function find I. 3. 4. Domain Range Section 22 Page 118
Domain Range Ex: Assume that is a function of,rewrite each equation using function notation, then find and 1. - * Increasing,Decreasing and Constant functions ^ ^ < v increasing. when v decreasing when v Constant when as x goes from left to right y goes up as x goes from left to right y goes down As x goes from left to - right y does not move Ex: The figure shows the graph of a function, Determine the intervals over which the function is increasing,decreasing or constant then find domain and range I. Increasing = Decreasing = Section 22 Page 119
Decreasing = Constant = Domain = Range = Increasing = Decreasing Constant = Domain = Range = Ex: If and find Ex: find the domain and range of the function f drown Domain= Section 22 Page 120
Domain= Range = Ex: Find the domain and range of Ex, Find the domain of the function Ex: Find the domain of Ex: If "find and write it in terms of Section 22 Page 121
Ex.. Identify the set of ordered pairs or the relation which defines as a function of x I. 3. 4. 5. Section 22 Page 122
Section 22 Page 123 4 Linear Functions Definition : A Linear function is a function that can be written in the form where. If Domain = Range = If Domain = Range = Ex: Graph ; give the domain and range Ex: Graph, give the domain and range Note: The graph of is a horizontal line passing through * Graphing a vertical line Ex: Graph ; give the domain and range of this relation
Section 22 Page 124 Note: The graph of is a vertical line passing through the point Ex: Write the equation of each Line in the graph it is -'z!? It is I,. If! + -2, Q: Write the equation of the y-axis The standard form of the equation of a Line where Ex: Graph and range ; give the domain
Section 22 Page 125 * Slope: is a numerical measure other steepness of. a line slope = geometrically Definition: the slope of a straight Line passing through two points and is where Note the slope of a line can be found only if the line is non vertical *the slope of a vertical line is undefined Ex find the slope of the line through the given points I. and and 3. and
Section 22 Page 126 3. and * zero slope The Slope of a horizontal line is zero Ex: find the slope of the line solution: Compare the slope by the coefficient of this is true always they are equal and -find the my-intercept Rule: If slope = m y-intercept= b. EX: Find the slope and yr-intercept of the following equation of a line slope y -intercept Ex: Graph the line passing through and having slope
Section 22 Page 127 Ex: Graph the line passing through and having slope Note. 1.rise = vertical movement + : move up : move down run z horizontal movement + : move right :move Left Ex: Graph
Section 22 Page 128 5 Equations of Lines: *Point slope form The Line with slope m passing through the point has an equation Ex: Find an equation of the line through having slope -2 * slope -intercept form The line with slope m and y-intercept b has an equation Ex: Find an equation of the line through and Ex: Use the graph of the Linear function shown here to find
Section 22 Page 129 a) slope, y -intercept, x-intercept b) Write the equation of Equations of vertical line and horizontal line 1. An equation of the vertical line through is An equation of the horizontal line through is Parallel and Perpendicular lines * Parallel lines Two distinct lines are parallel if and only if they have the same slope i.e. iff line 1 :Line 2 : slope of line one : slope of Line two
Section 22 Page 130 *Perpendicular lines: Two Lines, neither of which is vertical are perpendicular if and only if there slopes have a product (-1) i.e. iff So Ex: Fill in the table with the correct answer equation of the line slope of the parallel line slope of the perpendicular line Ex: Find the agnation in slope-intercept form of the line that passes through the point and satisfies: (a) parallel to the line (b) perpendicular to the line
Section 22 Page 131 Ex: Write an equation of a line 1. Through perpendicular to through parallel to 3. through perpendicular to 4. Through and perpendicular to Ex: Find K so that the line through and is (a) parallel to
Section 22 Page 132 (b) perpendicular to Ex: Determine whether are Collinear using slope Ex: Find the y-intercept of the line passing through the point to the line and perpendicular Ex: If the Line passing through the points and is parallel to the line "find
Section 22 Page 133 Ex: The line with I-intercept and y intercept intersects the line at the point find the value of.
Section 22 Page 134 6: Graphs of Basic Functions * Continuity (Informal definition) A Function is continuous over an interval of its domain if its hand drawn graph over that interval can be sketched without lifting the pencil from the paper -If a function is not continuous at a point then it has a discontinuity there Continuous at discontinuous at and Continuous at and discontinuous at 1 The Identity function \ is continuous on its entire domain The squaring function
Section 22 Page 135 Domain = Range = its decreasing on its increasing on its continuous on 3. The cubic function Domain= Range= increases on continuous on 4. The square root function Domain= Range= its increasing on its continuous on
Section 22 Page 136 5. The cubic root function Domain= Range= its increasing on its continuous on 6. The absolute value function Domain = Range = its increasing on its decreasing on its continuous on * Piecewise Defined functions Ex: Graph solution: If
Section 22 Page 137 If Ex: Graph If If Ex, Graph If If If
Section 22 Page 138 1. Ex: Give a rule for each piecewise-defined function and give the domain and range 3.
Section 22 Page 139 4. * Greatest integer Function the greatest integer function pairs every real number x with the greatest integer less than or equal to x In general if then for for for for for * Greatest integer function the graph Domain = Range =
Section 22 Page 140 the graph Range = is constant on the intervals It is discontinuous at all integer values in Ex: Graph first find the values of x such that then find a pattern for the length of each interval
Ex.. Graph Section 22 Page 141
Section 22 Page 142 Note? The length of each step in the greatest integer function step length = for Ex: graph
Section 22 Page 143 2-7 Graphing Techniques: * stretching and Shrinking 1.we will start by considering how the graph of to the graph of Exe Graph each function Compares Ex, Graph
Section 22 Page 144 How the graph of the graph of compares to the graph of Ex: Graph each function. 3. 4.
Ex: graph Section 22 Page 145
Section 22 Page 146 * Reflecting: Forming the mirror image of a graph across a line is called reflecting the graph across the line Ex: Graph * Translations: is a horizontal and vertical shifts Ex: Graph each function
Section 22 Page 147 Ex: Graph Ex: Graph
Ex: Graph Section 22 Page 148
Section 22 Page 149
Section 22 Page 150 Ex: Graph each function 1.
Section 22 Page 151 3. 4.
Section 22 Page 152 Ex: Find the intervals in which the function in the previous, example is 1. Increasing below the x-axis Ex: If the graph of the function is obtained from the graph of by means of: 1. a refection across the x-axis a horizontal shift 2 units to the left 3. a vertical shift 1 unit up find Ex: It the graph of 1. reflected across the y-axis shifted 1 unit to the left 3. shifted 3 units up find the new equation
Section 22 Page 153 Ex: If the adjacent figure is the graph of then find the domain n and Range R of the function
Section 22 Page 154 Sunday, December 09, 2012 7:29 PM 2-7 Graphing Techniques: * stretching and Shrinking 1.we will start by considering how the graph of to the graph of Exe Graph each function Compares 242 N It' 2 =it Ex, Graph
Section 22 Page 155! How the graph of the graph of compares to the graph of Ex: Graph each function. 3. 4. 12 N 1 1 Ital
Section 22 Page 156 1 1 Ital Inx Ex: graph * Reflecting: Forming the mirror image of a graph across a line is called reflecting the graph across the line Ex: Graph
Section 22 Page 157 * Translations: is a horizontal and vertical shifts Ex: Graph each function
Section 22 Page 158 Ex: Graph Ex: Graph
Ex: Graph Section 22 Page 159
Section 22 Page 160 a) (e)
Section 22 Page 161 Ex: Graph each function 1. :graph y=x~ - shift \ unit to the right 3reflection about x-axis 4. Shift 4 units up.
Section 22 Page 162 3. 4.
Section 22 Page 163 Ex: Find the intervals in which the function in the previous, example is 1. Increasing below the x-axis Ex: If the graph of the function is obtained from the graph of by means of: 1. a refection across the x-axis a horizontal shift 2 units to the left 3. a vertical shift 1 unit up find Ex: It the graph of 1. reflected across the y-axis
Section 22 Page 164 3. shifted 1 unit to the left shifted 3 units up find the new equation Ex: If the adjacent figure is the graph of then find the domain n and Range R of the function Ex: Test the symmetry with respect to the x-axis and y-axis 1.
Section 22 Page 165 3. 4. Ex: Are the following graphs symmetric about the origin 1.
Section 22 Page 166 Ex: Test for symmetry for the following graph, 1. 3. 4.
Section 22 Page 167 Notice the following important concepts regarding Symmetry :. A graph symmetric with respect to both the x- and y-axes is automatically symmetric with respect to the origin.. A graph symmetric with respect to the origin need not be symmetric with respect to either axis.. Of the three types of symmetry with respect to the x-axis, with respect to the y-axis, and with respect to the origin a graph possessing any two types must also exhibit the third type of symmetry. Ex: Decide whether each function is even, odd or neither 1. 3.
Section 22 Page 168 4. Ex: Test the symmetry of Ex: If the graph of the function
Section 22 Page 169 Ex: If is an even function such that find the coordinates of the two points that must he on the graph of
Section 22 Page 170 8 Function Operations and Composition Operations on functions Given two functions, then for all values of for which both are defined, the functions 1. 3. 4. are defined as follows Ex: Let and "find 1. 3. 4. Domain of a combination of two functions 1. Domain of 3. Domain = Domain = Ex: If
Section 22 Page 171 find and their domains Ex: If i find domain Ex: Use the given graph of functions G and g to evaluate I. 3. 4. The difference quotient Ex: Let find the difference quotient and simplify the expression
Section 22 Page 172 Composition of functions and domain If f and g are functions, then the composite function or composition of g and f is defined by we read it g after f Domain Domain Ex: Let find I. Ex: If and find I. and its domain and its donna,
Section 22 Page 173 Ex: If, find I. and its domain and its donna, Note: Ex: If find I. and its domain and its domain Ex: Find functions f and g such that
Section 22 Page 174 Ex: Find functions f and g such that 1. Ex: If Show that and Ex: If and the graph of the function has y-intercept 23 find m
Ex: Find the domain of the function Section 22 Page 175
Section 22 Page 176 3.1 Quadratic Functions and Models *polynomial Function A polynomial function of degree n; where n is a nonnegative integer is a function defined by an expression of the form where are real numbers with : is the leading coefficient is called the zero polynomial *Quadratic Functions a function f is a quadratic function if where a, b and c are real numbers the graph of a quadratic function is called Parabola axis vertex (maximum) point vertex (minimum) point open up axis open down Ex: Graph each function, give the domain and range 1.
Section 22 Page 177
Section 22 Page 178 Ex: Find the axis and vertex Domain, range, maximum or minimum point, x intercept and y-intercept of the parabola 1. by formula by completing squares
Section 22 Page 179 EX: Find two numbers whose sum is 12 and whose product is the maximum possible value Ex: Find a value C so that has exactly one X-intercept EX: For what values of a does has no x-intercept
Section 22 Page 180 Ex: Define a quadratic function having x intercepts 2 and 5 and y-intercept 5 Ex: find the closest point on the line to the point
Section 22 Page 181 Ex: A quadratic equation Its graph has vertex has a solution what is the other solution
3.2 synthetic division *Division algorithm Let and be polynomials with and of degree one or more of lower degree than there exists unique polynomials and such that where either. or the degree of is Less than the degree of. EX: Use Long division to divide EX! Use synthetic division to divide Section 22 Page 182
Section 22 Page 183 Note: To avoide errors use (0) as the coefficient for any missing terms, including a missing constant when setting a division Ex: Use synthetic division to divide Remainder theorem If the polynomial is divided by. then the remainder equal EX: let use the remainder theorem to find Ex: Decide whether the given number K is a zero of 1.
Section 22 Page 184 Ex: Decide whether the given number K is a zero of 1. 3. 4.
Section 22 Page 185 5. Ex: If is divided by, the remainder is 28 find K Ex: If (3) is a zero of find the other Zeros
Section 22 Page 186 Ex: If. Ex: If is divided by find the remainder
Section 22 Page 187 3.3 Zeros of Polynomial Functions Factor theorem: The polynomial if and only if Ex: Determine whether is a factor of : 1.
Section 22 Page 188 Ex: Factor factors if (5) is a zero of into Linear Rational zeros theorem: If is a rational number written in lowest terms, and if is a zero of ( a polynomial function with integer coefficients) then is a factor of the constant term and is a facts of the leading coefficient Ex: Do each of the following for the polynomial a) List all possible rational zeros b) Find all rational zeros and factor into Linear functions
Section 22 Page 189 Note: the rational zeros theorem gives only possible rational zeros it does not tell us whether these rational numbers are actual zeros or not Number of zeros: 1. Fundamental Theorem of Algebra Every function defined by a polynomial of degree (1) or more has at least one complex zero Number of zeros Theorem
Section 22 Page 190 A Function defined by a polynomial of degree (n) has at most (n) distinct zeros Note: the number of times a zero occurs is referred to as the multiplicity of the zero Ex: Find the zeros and their multiplicity for EX: Find a function defined by a polynomial of degree 3 that satisfies the given conditions a) Zeros : - 3,-2 and 5; b) 4 is a zero of multiplicity 3; Conjugate Zeros theorem: If defines a polynomial function having only real coefficients and
Section 22 Page 191 Conjugate Zeros theorem: If defines a polynomial function having only real coefficients and if is a zero of where a,b then is also a zero of Ex: Find a polynomial function coefficients and zeros: -4 and of least degree having only real Ex: Find all zeros of given that is a zero
Section 22 Page 192 Ex: Determine the possible number of positive real zeros and negative real zeros of Ex: factor. into Linear factors, given K is a zero of, (multiplicity 2)
Section 22 Page 193 Ex: Find all zeros and their multiplicity of EX: Factor factors of the constant term! factors of the leading coefficient possible zeros: EX: Find a polynomial of least degree having only real coefficients with
Section 22 Page 194 3.4 POLYNOMIAL FUNCTIONS OF HIGHER DEGREE All polynomial functions have graphs that are smooth continuous curves A smooth curve: no sharp corners A continuous curve: no breaks, holes, or gaps. continuous not continuous continuous smooth not polynomial not smooth polynomial polynomial not FAR-LEFT AND FAR RIGHT BEHAVIOR (End behavior) If P(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 (a n 0), then the leading term is
Ex1: Examine the leading term to determine the far-left and far-right behaviour of the graph: Section 22 Page 195
Section 22 Page 196 Def.: A turning point of a graph of a function is a point at which the graph changes from increasing to decreasing or vice versa Note: A polynomial function of degree n has at most n 1 turning points and at most n zeros. The intermediate value theorem (The Zero Location Theorem) Let P(x) be a polynomial function and let a and b be two distinct real numbers. If P(a) and P(b) have opposite signs, then there is at least one real number c between a and b such that P(c)=0
Section 22 Page 197 Ex: Use the Intermediate value theorem to verify that a) has a zero between 1 and b) has a zero between -1 and 0 Real Zeros of Polynomial Functions If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent. 1. c is a zero of f. c is a solution of the polynomial equation f (x) = 0. 3. x c is a factor of the polynomial f (x). 4. (c, 0) is an x-intercept of the graph of y = f (x). Even and odd Powers of (x-c) Theorem
Section 22 Page 198 If c is a real number and the polynomial function P(x) has (x-c) as a factor k times, then the graph of P will crosses the x-axis at (c, 0), if k=1 intersect but not cross the x-axis at (c, 0), provided k is an even positive integer.(the x-axis will be a tangent)(touches) cross the x-axis at (c, 0), and the x-axis is a tangent,provided k is an odd positive integer greater than 1. Ex3: Determine where the graph of the following polynomials crosses the x-axis and where the graph intersects but doesn't cross the x-axis and where it (crosses and makes a tangent) a)
Section 22 Page 199 b) A procedure for Graphing Polynomial Functions 1. 3. 4. 5. Determine the far-left and far-right behavior. Find the y-intercept. Find the x-intercept(s) and determine the behavior of the graph near the x-intercept(s). Find additional point on the graph. Check for symmetry. a. The graph of an even function is symmetric with respect to the y-axis. b. The graph of an odd function is symmetric with respect to the origin. 6. Use all the above information to sketch the graph Ex4: Sketch the graph of P (x) = 4x 2 x 4
Ex: Find a polynomial of least possible degree having the given graph : Section 22 Page 200
Section 22 Page 201 1) 2)