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Day 4 examples
Day 4 examplesjchartiersjsd
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This document contains a list of 5 coordinate pairs: (-x/2 - 1, y), (-1/2, 4), (-1/2, -1), (-3/2, 0), (-5/2, -1). The coordinate pairs appear to define points on the graph of a function and its inverse.Inverse Functions
Inverse FunctionsAlexander Burt
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The document discusses derivatives of functions and inverse functions. It defines a function and inverse function, and explains that a function must be one-to-one to have an inverse. It also states that the slope of the tangent line to the inverse function at a point (a,b) is the inverse of the slope of the tangent line to the original function at the same point (b,a).Tutorials--Functions and Their Inverses
Tutorials--Functions and Their InversesMedia4math
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This document provides 18 examples exploring the concept of functions and their inverses. It examines linear, quadratic, exponential, logarithmic, cubic, quartic and radical functions. Students are shown how the graph of y=x serves as the axis of reflection for determining a function's inverse, and that the inverse of a function is not always a function itself.Inverse functions
Inverse functionsJessica Garcia
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The document discusses inverse functions and how an inverse function undoes the operations of the original function. It provides examples of finding the inverse of functions by switching the x and y values and solving for y. The inverse of a function will be a function itself only if the original function passes the horizontal line test.
Day 8 examples u7w14
Day 8 examples u7w14jchartiersjsd
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This document contains 5 problems involving finding terms in expansions of polynomials. The problems involve finding specific terms that contain a given power of x in expansions of polynomials such as (3x^4 - 1)^9, (-x^3 + 2)^6, (x + 1)^3x, (x + 1)^x, and determining the value of m if one term in the expansion of (2x - m)^7 is -262500x^2y^5.Day 7 examples u7w14
Day 7 examples u7w14jchartiersjsd
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The document discusses binomial expansion, which is the process of multiplying out terms with two variables according to their power using the binomial theorem. It provides examples of expanding binomial expressions like (x + y)2, (x + y)3, and (x + y)4. It also notes that the sum of the exponents in each term equals the overall power, and the number of terms is always one more than the power. Finally, it provides the binomial theorem for expanding any binomial expression and finding a particular term.Day 4 examples u7w14
Day 4 examples u7w14jchartiersjsd
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This document discusses how to calculate arrangements when some items must be together or apart. It explains that when items need to be together, they should be counted as a single item to reduce the total items being arranged. Then the total number of arrangements is calculated by finding the total possible arrangements and subtracting the arrangements that do not satisfy the constraints of certain items being together or apart. Examples provided include arranging people in a row when some must or cannot sit together and arranging books on a shelf keeping books of each subject together.Day 3 examples u7w14
Day 3 examples u7w14jchartiersjsd
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Permutations refer to arrangements of objects in a definite order. Some key points:
- Permutations are represented by "nPn" where n is the total number of objects and r is the number being arranged.
- Permutations are used to calculate possibilities like license plates, phone numbers, and locker combinations.
- Restrictions like starting/ending conditions or requiring alternating arrangements reduce the number of possible permutations.
- Objects that are identical only count once toward the total number of permutations rather than being distinguishable.Day 2 examples u7w14
Day 2 examples u7w14jchartiersjsd
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Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.Day 1 examples u7w14
Day 1 examples u7w14jchartiersjsd
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This document provides examples and explanations of the fundamental counting principle and addition counting principle to solve combinatorics problems. It gives 8 examples of using the fundamental counting principle to count the number of possible outcomes of independent events. These include counting the number of volleyball shoe combinations, outfits that can be created from different clothing items, ways to select committees from groups of people, and 3-digit numbers with no repeating digits. It also provides 5 examples of using the addition counting principle to count outcomes when events are dependent, such as selecting a president and vice president of opposite sexes from a group.
Day 7 examples u6w14
Day 7 examples u6w14jchartiersjsd
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This document discusses graphing composite functions. It provides examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f(x) and g(x), sketching the graphs of the composite functions, and stating their domains. It also gives examples of determining possible functions f(x) and g(x) that satisfy given composite functions.Day 5 examples u6w14
Day 5 examples u6w14jchartiersjsd
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1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.Day 4 examples u6w14
Day 4 examples u6w14jchartiersjsd
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The graph is a linear function with a domain of all real numbers and a range of real numbers greater than or equal to 3. The graph is a line with a y-intercept of 3 that increases at a rate of 1 as x increases.Day 3 examples u6w14
Day 3 examples u6w14jchartiersjsd
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Rational functions are functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. For example, comparing rational functions like 2x/(x^2 - 4) and (x-1)/(x+1). Horizontal asymptotes of rational functions occur when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Day 1 examples u6w14
Day 1 examples u6w14jchartiersjsd
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This document discusses how to find the sum, difference, product, and quotient of functions. The sum of functions is found by adding the y-coordinates of each function. The difference is found by subtracting the y-coordinates. The product is represented as h(x) = f(x)g(x) and the quotient is represented as h(x) = f(x)/g(x). Examples are provided for adding and subtracting functions.Mental math test outline
Mental math test outlinejchartiersjsd
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The document outlines a mental math test covering polynomials. It includes short answer questions testing long division, synthetic division, the remainder theorem, and finding the degree, leading coefficient, and y-intercept of polynomials. The test also covers matching graphs to polynomial equations and word problems involving fully factoring polynomials and two graphs. Multiple choice questions will require explaining solutions, while long answer questions involve fully factoring polynomials and word problems.Day 8 examples u5w14
Day 8 examples u5w14jchartiersjsd
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The document contains two polynomial word problems. The first asks to write a function V(x) to express the volume of a box with dimensions x, x+2, x+10 in terms of x, and find possible x values if the volume is 96 cm^3. The second problem describes a block of ice that is initially 3 ft by 4 ft by 5 ft, and asks to write a function to model reducing each dimension by the same amount to reach a volume of 24 ft^3, and determine how much to remove from each dimension.Day 7 examples u5w14
Day 7 examples u5w14jchartiersjsd
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The document provides 3 polynomial word problems: 1) finding the equation for a polynomial given its graph f(x) = -(x - 2)2(x + 1), 2) determining the polynomial P(x) when divided by (x - 3) with a quotient of 2x^2 + x - 6 and remainder of 4, and 3) finding the value of a if (x - 2) is a factor of ax^3 + 4x^2 + x - 2. It also gives a 4th problem of determining the value of k when 2x^3 + kx^2 - 3x + 2 is divided by x - 2 with a remainder of 4.Day 5 examples u5w14
Day 5 examples u5w14jchartiersjsd
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Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.Day 3 examples u5w14
Day 3 examples u5w14jchartiersjsd
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This document asks which of the following binomials are factors of the expression 9x - 12. The options given are x + 3 and x - 2. Of these two options, x - 2 is a factor of 9x - 12, since (3x - 4)(3x - 4) = 9x^2 - 12x + 12 = 9x - 12.Day 2 examples u5w14
Day 2 examples u5w14jchartiersjsd
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Synthetic division is a method for dividing a polynomial by a linear factor such as (x - a). The coefficients of the polynomial are written in a column. The divisor is written above the column. Working from right to left, each term of the divisor is multiplied by the coefficient above and subtracted from the number above. The result is the quotient and remainder of dividing the polynomial by the linear factor.Day 1 examples u5w14
Day 1 examples u5w14jchartiersjsd
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A polynomial function contains terms of variables raised to whole number powers that are added or subtracted. Examples of polynomials are x^2 + 5x + 3 and 3x^4 - 2x^3 + x^2 - 4. Long division can be used to divide polynomials, with the process being similar to dividing numbers. The result of polynomial division is the quotient polynomial and a remainder.Day 10 examples u4w14
Day 10 examples u4w14jchartiersjsd
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This document discusses solving trigonometric equations with identities. It provides three example trig equations to solve for the variable x: 2sinx - 14 = 0, 1 - sin^2x = 3cosx - 2, and cosxsin2x - 2sinx = -2. It also gives the general solution of 4cos(2θ) + 2 = 0 to solve for θ.Day 11 examples u4w14
Day 11 examples u4w14jchartiersjsd
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This document discusses verifying trigonometric identities and determining non-permissible values for identities. It first verifies two identities by showing that = and = 30° are solutions. It then identifies the non-permissible values of θ for the identity tan θ = 1/cot θ on the interval from 0 to 2π as being θ = nπ where n is any integer.More Related Content
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Day 4 examples u7w14
Day 4 examples u7w14jchartiersjsd
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This document discusses how to calculate arrangements when some items must be together or apart. It explains that when items need to be together, they should be counted as a single item to reduce the total items being arranged. Then the total number of arrangements is calculated by finding the total possible arrangements and subtracting the arrangements that do not satisfy the constraints of certain items being together or apart. Examples provided include arranging people in a row when some must or cannot sit together and arranging books on a shelf keeping books of each subject together.Day 3 examples u7w14
Day 3 examples u7w14jchartiersjsd
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Permutations refer to arrangements of objects in a definite order. Some key points:
- Permutations are represented by "nPn" where n is the total number of objects and r is the number being arranged.
- Permutations are used to calculate possibilities like license plates, phone numbers, and locker combinations.
- Restrictions like starting/ending conditions or requiring alternating arrangements reduce the number of possible permutations.
- Objects that are identical only count once toward the total number of permutations rather than being distinguishable.Day 2 examples u7w14
Day 2 examples u7w14jchartiersjsd
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Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.Day 1 examples u7w14
Day 1 examples u7w14jchartiersjsd
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This document provides examples and explanations of the fundamental counting principle and addition counting principle to solve combinatorics problems. It gives 8 examples of using the fundamental counting principle to count the number of possible outcomes of independent events. These include counting the number of volleyball shoe combinations, outfits that can be created from different clothing items, ways to select committees from groups of people, and 3-digit numbers with no repeating digits. It also provides 5 examples of using the addition counting principle to count outcomes when events are dependent, such as selecting a president and vice president of opposite sexes from a group.Day 7 examples u6w14
Day 7 examples u6w14jchartiersjsd
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This document discusses graphing composite functions. It provides examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f(x) and g(x), sketching the graphs of the composite functions, and stating their domains. It also gives examples of determining possible functions f(x) and g(x) that satisfy given composite functions.Day 5 examples u6w14
Day 5 examples u6w14jchartiersjsd
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1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.Day 4 examples u6w14
Day 4 examples u6w14jchartiersjsd
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The graph is a linear function with a domain of all real numbers and a range of real numbers greater than or equal to 3. The graph is a line with a y-intercept of 3 that increases at a rate of 1 as x increases.Day 3 examples u6w14
Day 3 examples u6w14jchartiersjsd
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Rational functions are functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. For example, comparing rational functions like 2x/(x^2 - 4) and (x-1)/(x+1). Horizontal asymptotes of rational functions occur when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.Day 1 examples u6w14
Day 1 examples u6w14jchartiersjsd
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This document discusses how to find the sum, difference, product, and quotient of functions. The sum of functions is found by adding the y-coordinates of each function. The difference is found by subtracting the y-coordinates. The product is represented as h(x) = f(x)g(x) and the quotient is represented as h(x) = f(x)/g(x). Examples are provided for adding and subtracting functions.Mental math test outline
Mental math test outlinejchartiersjsd
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The document outlines a mental math test covering polynomials. It includes short answer questions testing long division, synthetic division, the remainder theorem, and finding the degree, leading coefficient, and y-intercept of polynomials. The test also covers matching graphs to polynomial equations and word problems involving fully factoring polynomials and two graphs. Multiple choice questions will require explaining solutions, while long answer questions involve fully factoring polynomials and word problems.Day 8 examples u5w14
Day 8 examples u5w14jchartiersjsd
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The document contains two polynomial word problems. The first asks to write a function V(x) to express the volume of a box with dimensions x, x+2, x+10 in terms of x, and find possible x values if the volume is 96 cm^3. The second problem describes a block of ice that is initially 3 ft by 4 ft by 5 ft, and asks to write a function to model reducing each dimension by the same amount to reach a volume of 24 ft^3, and determine how much to remove from each dimension.Day 7 examples u5w14
Day 7 examples u5w14jchartiersjsd
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The document provides 3 polynomial word problems: 1) finding the equation for a polynomial given its graph f(x) = -(x - 2)2(x + 1), 2) determining the polynomial P(x) when divided by (x - 3) with a quotient of 2x^2 + x - 6 and remainder of 4, and 3) finding the value of a if (x - 2) is a factor of ax^3 + 4x^2 + x - 2. It also gives a 4th problem of determining the value of k when 2x^3 + kx^2 - 3x + 2 is divided by x - 2 with a remainder of 4.Day 5 examples u5w14
Day 5 examples u5w14jchartiersjsd
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Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.Day 3 examples u5w14
Day 3 examples u5w14jchartiersjsd
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This document asks which of the following binomials are factors of the expression 9x - 12. The options given are x + 3 and x - 2. Of these two options, x - 2 is a factor of 9x - 12, since (3x - 4)(3x - 4) = 9x^2 - 12x + 12 = 9x - 12.Day 2 examples u5w14
Day 2 examples u5w14jchartiersjsd
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Synthetic division is a method for dividing a polynomial by a linear factor such as (x - a). The coefficients of the polynomial are written in a column. The divisor is written above the column. Working from right to left, each term of the divisor is multiplied by the coefficient above and subtracted from the number above. The result is the quotient and remainder of dividing the polynomial by the linear factor.Day 1 examples u5w14
Day 1 examples u5w14jchartiersjsd
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A polynomial function contains terms of variables raised to whole number powers that are added or subtracted. Examples of polynomials are x^2 + 5x + 3 and 3x^4 - 2x^3 + x^2 - 4. Long division can be used to divide polynomials, with the process being similar to dividing numbers. The result of polynomial division is the quotient polynomial and a remainder.Day 10 examples u4w14
Day 10 examples u4w14jchartiersjsd
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This document discusses solving trigonometric equations with identities. It provides three example trig equations to solve for the variable x: 2sinx - 14 = 0, 1 - sin^2x = 3cosx - 2, and cosxsin2x - 2sinx = -2. It also gives the general solution of 4cos(2θ) + 2 = 0 to solve for θ.Day 11 examples u4w14
Day 11 examples u4w14jchartiersjsd
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This document discusses verifying trigonometric identities and determining non-permissible values for identities. It first verifies two identities by showing that = and = 30° are solutions. It then identifies the non-permissible values of θ for the identity tan θ = 1/cot θ on the interval from 0 to 2π as being θ = nπ where n is any integer.