Grade 12 Advanced Functions
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In this interactive course, you will learn Grade 12 Advanced Functions at your own pace. By the end you will feel confident, know how to ace your exams at school, and be able to apply advanced mathematics to everyday problem solving.
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Lesson 1: Rates of change (how to represent rates of change as well as real-world examples)
Quiz: Lesson 1
Mini Challenge: Lesson 1
Worksheet: Lesson 1
Lesson 2: Explore situations in which the rate of change is zero, constant, or changing (include real world applications)
Quiz: Lesson 2
Mini Challenge: Lesson 2
Worksheet: Lesson 2
Lesson 3: Calculating rates of change for various functions (linear, quadratic, sinusoidal etc)
Worked Out Example: Lesson 3
Quiz: Lesson 3
Mini Challenge: Lesson 3
Worksheet: lesson 3
Lesson 4: Average rate of change vs. Instantaneous rate of change
Quiz: Lesson 4
Mini Challenge: Lesson 4
Worksheet: Lesson 4
Lesson 5: Review: What are tangent lines and what does "normal to the tangent" mean
Quiz: Lesson 5
Game Time! Select the correct answer
Game Time! Guess the Answer
Mini Challenge: Lesson 5
Worksheet: Lesson 5
Lesson 6: Making connections between the slope of a secant on the graph of a function and the average rate of change
Quiz: Lesson 6
Mini Challenge: Lesson 6
Worksheet: Lesson 6
Lesson 7: Making connections between the slope of a tangent to a point on a graph of a function and the instantaneous rate of change of a function at that point
Quiz: Lesson 7
Mini Challenge: Lesson 7
Worksheet: Lesson 7
Lesson 8: Determine the approximate slope of the tangent to a given point on the graph of a function by using the slopes of secants through the given point and make connections to average and instantaneous rates of change
Quiz: Lesson 8
Mini Challenge: Lesson 8
Worksheet: Lesson 8
Lesson 9: Solve problems involving average and instantaneous rates of change, including problems arising from real-world applications (can use tech like desmos)
Quiz: Lesson 9
Game Time! Drag the words (L6 - 9)
Mini Challenge: Lesson 9
Worksheet: lesson 9
Lesson 10: What is the limit of a function?
Quiz: Lesson 10
Mini Challenge: Lesson 10
Worksheet: lesson 10
Lesson 11: Key features (domain, range, max, min, etc) of the graphs of functions created by adding,subtracting,multiplying,or dividing functions( [e.g., f(x) = 2-x sin 4x, g(x) = x2 + 2x , h(x) =sin x/cos x & describe factors that affect these properties
Quiz: Lesson 11
Mini Challenge: Lesson 11
Worksheet: lesson 11
Lesson 12: Real-world applications of combinations of functions
Quiz: Lesson 12
Game Time! Flip and learn (L11 - 12)
Mini Challenge: Lesson 12
Worksheet: lesson 12
Lesson 13: Explain properties (i.e., odd, even, or neither; increasing/decreasing behaviours) of functions formed by adding, subtracting, multiplying, and dividing general functions [e.g., f(x) + g(x), f(x)g(x)]
Quiz: Lesson 13
Game Time! Even or odd? (L13)
Mini Challenge: Lesson 13
Worksheet: Lesson 13
Lesson 14: Composite functions / Composition of functions, interpret the notation of composite functions
Worked Out Example: Lesson 14
Quiz: Lesson 14
Mini Challenge: Lesson 14
Worksheet: Lesson 14
Lesson 15: Composition of two functions: real world applications
Quiz: Lesson 15
Mini Challenge: Lesson 15
Worksheet: Lesson 15
Lesson 16: Algebraically determine the composition of two functions
Quiz: Lesson 16
Game Time! Drag and drop (L14 - 16)
Mini Challenge: Lesson 16
Worksheet: Lesson 16
Lesson 17: Understand how to represent complex functions as composites of simpler functions
Worked Out Example: Lesson 17
Quiz: Lesson 17
Mini Challenge: Lesson 17
Worksheet: lesson 17
Lesson 18: Demonstrate, by giving examples for functions represented in a variety of ways
Quiz: Lesson 18
Mini Challenge: Lesson 18
Worksheet: Lesson 18
Lesson 19: Make connections, through investigation using technology, between transformations
Quiz: Lesson 19
Game Time! Fill in the blanks (L17 - 19)
Game Time! Identify the transformations (L19)
Mini Challenge: Lesson 19
Worksheet: Lesson 19
Lesson 20: Compare various functions with one another (polynomial, rational, trigonometric, exponential, logarithmic)
Mini Challenge: Lesson 20
Worksheet: Lesson 20
Lesson 21: solve graphically and numerically equations and inequalities whose solutions are not accessible by standard algebraic techniques (can use desmos or GDC)
Quiz: Lesson 21
Mini Challenge: Lesson 21
Worksheet: Lesson 21
Lesson 22: locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change
Quiz: Lesson 22
Game Time! Find the correct solutions (L21 - 22)
Mini Challenge: Lesson 22
Game Time! Crossword
Worksheet: Lesson 22
Quiz: Lesson 1
Quiz: Lesson 1 Part 2
Mini Challenge: Lesson 1
Mini Challenge: Lesson 1 Part 2
Worksheet: Lesson 1
Lesson 2: Finding the degree from the equation in expanded or factored form
Worked Out Example: Lesson 2
Quiz: Lesson 2
Mini Challenge: Lesson 2
Worksheet: Lesson 2
Lesson 3: Finding the leading coefficient from the equation in expanded or factored form
Quiz: Lesson 3
Game Time! Is it a polynomial?
Mini Challenge: Lesson 3
Worksheet: Lesson 3
Lesson 4: Comparing (using graphing technology) the numeric, graphical, and algebraic representations of polynomial (i.e., linear, quadratic, cubic, quartic) functions
Quiz: Lesson 4
Mini Challenge: Lesson 4
Worksheet: lesson 4
Quiz: Lesson 5
Game Time! Flip and learn
Mini Challenge: Lesson 5
Worksheet: Lesson 5
Lesson 7: Making connections (using graphing technology) between a polynomial function given in factored form [e.g., f(x) = 2(x – 3)(x + 2)(x – 1)] and the x-intercepts of its graph
Worked Out Example: Lesson 7
Quiz: Lesson 6/7
Mini Challenge: Lesson 6/7
Worksheet: Lesson 6/7
Lesson 8: Sketching the graph of a polynomial function given in factored form using its key features.
Quiz: Lesson 8
Game Time! Select the right graph
Mini Challenge: Lesson 8
Worksheet: Lesson 8
Lesson 9: Determining (using technology, desmos is fine) the roles of the parameters a, k, d, and c in functions of the form y = af(k(x – d)) + c, and describe these roles in terms of transformations on the graphs of f(x) = x and f(x) = x
Quiz: Lesson 9
Game Time! Put the graphs in the correct order
Mini Challenge: Lesson 9
Worksheet: Lesson 9
Lesson 10: Determining an equation of a polynomial function that satisfies a given set of conditions (In the lesson include that there may be more than one polynomial function that can satisfy a given set of conditions)
Worked Out Example: Lesson 10
Quiz: Lesson 10
Mini Challenge: Lesson 10
Worksheet: lesson 10
Worked Out Example: Lesson 11
Quiz: Lesson 11
Mini Challenge: Lesson 11
Worksheet: Lesson 11
Lesson 12: Determining and comparing the properties of even and odd polynomial functions, and determine whether a given polynomial function is even, odd, or neither
Worked Out Example: Lesson 12
Quiz: Lesson 12
Game Time! Drag and drop
Mini Challenge: Lesson 12
Worksheet: Lesson 12
Lesson 13: Determining (with and without technology) key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals)
Worked Out Example: Lesson 13
Quiz: Lesson 13
Mini Challenge: Lesson 13
Worksheet: Lesson 13
Lesson 14: Determining (with and without technology) key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals)
Quiz: Lesson 14
Mini Challenge: Lesson 14
Worksheet: Lesson 14
Lesson 15: Sketching the graph of a simple rational function using its key features, given the algebraic representation of the function
Worked Out Example: Lesson 15
Quiz: Lesson 15
Mini Challenge: Lesson 15
Worksheet: lesson 15
Lesson 16: Understanding the connection between vertical asymptotes and limits that equal to ∞
Quiz: Lesson 16
Mini Challenge: Lesson 16
Worksheet: Lesson 16
Lesson 17: Given the equation of a rational function, determine the equation of the horizontal asymptote
Quiz: Lesson 17
Mini Challenge: Lesson 17
Worksheet: Lesson 17
Lesson 19: Given the equation of a polynomial or rational function, draw an accurate sketch showing all key features
Quiz: Lesson 18/19
Game Time! Question set
Mini Challenge: Lesson 18/19
Worksheet: Lesson 19
Lesson 20: Performing long division of polynomials, perform synthetic division of polynomials, state the restrictions, and write the division statement
Worked Out Example: Lesson 20
Quiz: Lesson 20
Mini Challenge: Lesson 20
Worksheet: lesson 20
Lesson 20B: Performing long division of polynomials, perform synthetic division of polynomials, state the restrictions, and write the division statement
Worked Out Example: Lesson 20B
Quiz: Lesson 20 B
Game Time! Drag and drop
Mini Challenge: Lesson 20B
Worksheet: Lesson 20B
Lesson 21: Making connections (using technology) between the polynomial function f(x), the divisor x – a, the remainder from the division f(x)/x-a, and f(a) to verify the remainder theorem and the factor theorem
Quiz: Lesson 21
Mini Challenge: Lesson 21
Worksheet: lesson 21
Lesson 22: Using the factor theorem to determine whether a binomial is a factor of a higher-order polynomial
Worked Out Example: Lesson 22
Quiz: Lesson 22
Game Time! Find the factors
Mini Challenge: Lesson 22
Worksheet: Lesson 22
Lesson 23: Factoring polynomial expressions in one variable, of degree no higher than four, by selecting and applying strategies (i.e., common factoring, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem)
Worked Out Example: Lesson 23
Quiz: Lesson 23
Mini Challenge: Lesson 23
Worksheet: Lesson 23
Lesson 24: Determining (using technology) the connection between the real roots of a polynomial equation and the x-intercepts of the graph of the corresponding polynomial function, and describe this connection
Worked Out Example: Lesson 24
Quiz: Lesson 24
Mini Challenge: Lesson 24
Worksheet: Lesson 24
Lesson 26: Solving polynomial equations in one variable, of degree no higher than four (e.g., 2x3 – 3x2 + 8x – 12 = 0), by selecting and applying strategies
Worked Out Example: Lesson 25-26
Quiz: Lesson 26
Game Time! Drag and drop
Mini Challenge: Lesson 26
Worksheet: Lesson 26
Lesson 27: Determining (using technology) the connection between the real roots of a rational equation and the x-intercepts of the graph of the corresponding rational function, and describe this connection
Quiz: Lesson 27
Mini Challenge: Lesson 27
Worksheet: Lesson 27
Lesson 28: Solving simple rational equations in one variable algebraically, and verify solutions using technology
Worked Out Example: Lesson 28
Quiz: Lesson 28
Game Time! Flip and learn
Mini Challenge: Lesson 28
Worksheet: Lesson 28
Lesson 29: Solving problems involving applications of polynomial and simple rational functions and equations (In the lesson include problems involving the factor theorem or remainder theorem, such as determining the values of k for which the function f(x)
Worked Out Example: Lesson 29
Quiz: Lesson 29
Game Time! Guess the Answer
Mini Challenge: Lesson 29
Worksheet: Lesson 29
Lesson 30: Explaining, for polynomial and simple rational functions, the difference between the solution to an equation in one variable and the solution to an inequality in one variable, and demonstrate that given solutions satisfy an inequality
Quiz: Lesson 30
Mini Challenge: Lesson 30
Worksheet: lesson 30
Lesson 31: Determining solutions to polynomial inequalities in one variable & to simple rational inequalities in one variable by graphing the corresponding functions,using graphing technology, & identifying intervals for which x satisfies the inequalities
Worked Out Example: Lesson 31
Quiz: Lesson 31
Mini Challenge: Lesson 31
Worksheet: Lesson 31
Lesson 32: Solving factorable polynomial inequalities algebraically and by graphing
Worked Out Example: Lesson 32
Quiz: Lesson 32
Game Time! Question set
Mini Challenge: Lesson 32
Worksheet: Lesson 32
Lesson 33: Given the equation of a higher-order polynomial: - Predict the behaviour of the graph near the x-intercepts - Predict the end behaviour - Determine the maximum number of turning points
Quiz: Lesson 33
Mini Challenge: Lesson 33
Worksheet: Lesson 33
Lesson 34: Drawing a detailed sketch illustrating the main features from previous lesson
Worked Out Example: Lesson 34
Quiz: Lesson 34
Game Time! Fill in the blanks
Mini Challenge: Lesson 34
Worksheet: Lesson 34
Lesson 35: Solving linear inequalities and factorable polynomial inequalities in one variable and represent the solutions on a number line and algebraically
Worked Out Example: Lesson 35
Quiz: Lesson 35
Mini Challenge: Lesson 35
Game Time! Drag and Drop
Worksheet: Lesson 35
Lesson 1: Recognizing the logarithm of a number to a given base as the exponent to which the base must be raised to get the number
Worked Out Example: Lesson 1
Quiz: Lesson 1
Mini Challenge: Lesson 1
Worksheet: Lesson 1
Lesson 2: Recognizing the operation of finding the logarithm to be the inverse operation
Worked Out Example: Lesson 2
Quiz: Lesson 2
Mini Challenge: Lesson 2
Worksheet: Lesson 2
Lesson 3: Determining (with technology) the approximate logarithm of a number to any base, including base 10
Worked Out Example: Lesson 3
Quiz: Lesson 3
Mini Challenge: Lesson 3
Worksheet: Lesson 3
Quiz: Lesson 4
Mini Challenge: Lesson 4
Worksheet: Lesson 4
Lesson 5: Solving simple exponential equations by rewriting them in logarithmic form
Worked Out Example: Lesson 5
Quiz: Lesson 5
Game Time! Question set
Mini Challenge: Lesson 5
Worksheet: lesson 5
Lesson 7: Making connections between laws of exponents and the laws of logarithms and verify the laws of logarithms with or without technology
Quiz: Lesson 7
Mini Challenge: Lesson 7
Worksheet: Lesson 7
Lesson 8: Using the laws of logarithms to simplify and evaluate numerical expressions
Worked Out Example: Lesson 8
Quiz: Lesson 8
Mini Challenge: Lesson 8
Worksheet: Lesson 8
Lesson 9: Laws of logs: PROOFS (log ab = log a + log b), (log a/b = log a- log b), (log a^x = x log a)
Quiz: Lesson 9
Mini Challenge: Lesson 9
Worksheet: Lesson 9
Quiz: Lesson 10
Game Time! Fill in the blanks
Mini Challenge: Lesson 10
Worksheet: Lesson 10
Lesson 11: Determining (with and without technology) key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, increasing/decreasing behaviour)
Worked Out Example: Lesson 11
Quiz: Lesson 11
Mini Challenge: Lesson 11
Worksheet: Lesson 11
Lesson 12: Using logarithms to transform a given relationship to linear form, and determine unknown constants by considering the gradient and/or intercept
Quiz: Lesson 12
Mini Challenge: Lesson 12
Worksheet: Lesson 12
Lesson 13: The relationship between an exponential function and the corresponding logarithmic function to be that of a function and its inverse
Quiz: Lesson 13
Mini Challenge: Lesson 13
Worksheet: lesson 13
Worksheet: Lesson 14/2 MERGED
Quiz: Lesson 14/2 MERGED
Game Time! Inverse Functions
Mini Challenge: Lesson 14/2 MERGED
Lesson 15: Determining the roles of the parameters d and c in functions of the form y=log10(x–d)+c & the roles of the parameters a & k in functions of the form y=alog10(kx), & describe these roles in terms of transformations on the graph of f(x)=log1
Quiz: Lesson 15
Game Time! Question set
Game Time! Find the right graph
Game Time! Find the right graph
Game Time! Find the right graph
Game Time! Find the right graph
Mini Challenge: Lesson 15
Worksheet: Lesson 15
Lesson 16/17: Explore Ln xLesson 18: Finding the equation of the inverse given the equation of an exponential or logarithmic function
Quiz: Lesson 16/17
Mini Challenge: Lesson 16/17
Worksheet: Lesson 16/17
Lesson 18: Finding the equation of the inverse given the equation of an exponential or logarithmic function
Worked Out Example: Lesson 18
Quiz: Lesson 18
Game Time! Question set
Mini Challenge: Lesson 18
Worksheet: Lesson 18
Lesson 19: Posing problems based on real-world applications of exponential and logarithmic functions and solve these problems by using a given graph or a graph generated with technology from a table of values or from its equation
Worked Out Example: Lesson 19
Quiz: Lesson 19
Game Time! Drag and drop
Mini Challenge: Lesson 19
Worksheet: Lesson 19
Lesson 20: Recognizing equivalent algebraic expressions involving logarithms and exponents, and simplify expressions of these types
Quiz: Lesson 20
Mini Challenge: Lesson 20
Worksheet: Lesson 20
Lesson 22: Solving simple logarithmic equations in one variable algebraically
Worked Out Example: Lesson 22
Quiz: Lesson 22
Mini Challenge: Lesson 22
Worksheet; Lesson 22
Lesson 23: Solving logarithmic equations, including checking for inadmissible solutions that would give a negative base or argument
Quiz: Lesson 23
Game Time! Fill in the blanks
Mini Challenge: Lesson 23
Worksheet: Lesson 23
Lesson 25: Understanding how logarithmic scales are used to measure earthquakes, sound, and acidity
Quiz: Lesson 25
Game Time! Question set
Mini Challenge: Lesson 25
Game Time! Flip and Learn
Worksheet: Lesson 25
Lesson 1: Defining the radian measure of an angle as the length of the arc that subtends this angle at the centre of a unit circle, and develop and apply the relationship between radian and degree measure
Quiz: Lesson 1
Mini Challenge: Lesson 1
Worksheet: Lesson 1
Lesson 2: Representing radian measure in terms of π and as a rational number
Quiz: Lesson 2
Mini Challenge: Lesson 2
Worksheet: Lesson 2
Lesson 3: Determining, with technology, the primary trigonometric ratios (i.e., sine, cosine, tangent) and the reciprocal trigonometric ratios (i.e., cosecant, secant, cotangent) of angles expressed in radian measure
Quiz: Lesson 3
Mini Challenge: Lesson 3
Worksheet: Lesson 3
Lesson 4: Determining, without technology, the exact values of the primary trigonometric ratios and the reciprocal trigonometric ratios for the special angles 0, π/6, π/4, π/3, π/2, and their multiples less than or equal to 2π
Worked Out Example: Lesson 4
Quiz: Lesson 4
Mini Challenge: Lesson 4
Worksheet: Lesson 4
Lesson 5: Predicting the signs of trig ratios, given the quadrant
Quiz: Lesson 5
Game Time! Question set
Mini Challenge: Lesson 5
Worksheet: Lesson 5
Lesson 6: Understanding the concept of periodic graphs
Quiz: Lesson 6
Mini Challenge: Lesson 6
Worksheet: Lesson 6
Lesson 7: Sketching the graphs of f(x) = sin x and f(x) = cos x for angle measures expressed in radians, and determine and describe some key properties in terms of radians
Quiz: Lesson 7
Mini Challenge: Lesson 7
Worksheet: Lesson 7
Lesson 8: Identifying the features (including period, amplitude, principal axis) of the basic trig graphs, including y = tan θ
Quiz: Lesson 8
Mini Challenge: Lesson 8
Worksheet: Lesson 8
Lesson 9: Understanding the connection of the vertical asymptotes in y = tan θ to the x, y, r definitions
Lesson 10: Using the basic trig graphs to determine the values of some trig ratios
Quiz: Lesson 9/10
Mini Challenge: Lesson 9/10
Worksheet: Lesson 9/10
Lesson 11: Making connections between the tangent ratio and the tangent function by using technology to graph the relationship between angles in radians and their tangent ratios (continued in next lesson)
Quiz: Lesson 11
Game Time! Fill in the blanks
Mini Challenge: Lesson 11
Worksheet: Lesson 11
Lesson 13: Making connections between the equations, graphs, and features of transformed trig functions
Quiz: Lesson 12/13
Mini Challenge: Lesson 12/13
Worksheet: lesson 12/13
Lesson 14: Graphing, with technology and using the primary trigonometric functions, the reciprocal trigonometric functions (i.e., cosecant, secant, cotangent) for angle measures expressed in radians
Quiz: Lesson 14
Mini Challenge: Lesson 14
Worksheet: Lesson 14
Lesson 15: Determining and describing key properties of the reciprocal functions
Quiz: Lesson 15
Game Time! Question set
Mini Challenge: Lesson 15
Worksheet: lesson 15
Lesson 16: Determining the amplitude, period, and phase shift of sinusoidal functions whose equations are given in the form f(x) = a sin (k(x – d)) + c or f(x) = a cos(k(x – d)) + c, with angles expressed in radians
Quiz: Lesson 16
Mini Challenge: Lesson 16
Worksheet: Lesson 16
Lesson 17: Sketching graphs of y = a sin (k(x–d)) + c a & y = a cos(k(x–d))+c by applying transformations to the graphs of f(x)=sin x & f(x)=cos x with angles expressed in radians, & state the period, amplitude, & phase shift of the transformed function
Worked Out Example: Lesson 17
Quiz: Lesson 17
Game Time! Drag and drop
Mini Challenge: Lesson 17
Worksheet: lesson 17
Lesson 18: Representing a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians
Worked Out Example: Lesson 18
Quiz: Lesson 18
Game Time! Question set
Mini Challenge: Lesson 18
Worksheet: Lesson 18
Lesson 19: Posing problems based on applications involving a trigonometric function with domain expressed in radians & solve these problems by using a given graph or a graph generated with or without technology from a table of values or from its equation
Quiz: Lesson 19
Game Time! Find the right trig functions
Mini Challenge: Lesson 19
Worksheet: Lesson 19
Quiz: Lesson 20/21
Lesson 20/21: Using special triangles to find the exact values of trig ratios for 30°, 45°, and 60°
Mini Challenge: Lesson 20/21
Worksheet: Lesson 20/21
Lesson 22: Solving simple trig equations with a calculator but without using graphing technology
Quiz: Lesson 22
Mini Challenge: Lesson 22
Worksheet: Lesson 22
Lesson 24: Finding all the solutions of simple trigonometric equations lying in a specified interval (general forms of solution are not included)
Quiz: Lesson 23/24
Game Time! Drag and drop
Mini Challenge: Lesson 23/24
Worksheet: Lesson 23/24
Lesson 25/26: Exploring the algebraic development of the compound angle formulas and use the formulas to determine exact values of trigonometric ratios
Quiz: Lesson 25/26
Game Time! Drag and drop
Game Time! Dialog Cards
Mini Challenge: Lesson 25/26
Worksheet: lesson 25/26
Lesson 27: Recognize that trigonometric identities are equations that are true for every value in the domain
Quiz: Lesson 27
Mini Challenge: Lesson 27
Worksheet: Lesson 27
Lesson 28: Using trig identities for the simplification and exact evaluation of expressions
Worked Out Example: Lesson 28
Quiz: Lesson 28
Mini Challenge: Lesson 28
Worksheet: lesson 28
Lesson 29: Selecting an identity or identies appropriate to the context (Include: Sec2θ = 1 + tan2θ and cosec2θ = 1 + cot2θ, The expansions of sin (A+ B), cos(A+ B) and tan(A+ B), The formulae for sin2A, cos2A and tan2A , The expression of a sinθ + b cosθ
Quiz: Lesson 29
Game Time! True or false
Game Time! Find the correct solution
Mini Challenge: Lesson 29
Worksheet: Lesson 29
Lesson 30: Solving linear and quadratic trigonometric equations, with and without graphing technology, for the domain of real values from 0 to 2π, and solve related problems (could be split into two lessons if too much)
Quiz: Lesson 30
Game Time! Drag and drop
Mini Challenge: Lesson 30
Game Time! Memory Game
Worksheet: Lesson 30
Let's Tinker!
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