This document discusses trigonometric functions such as sine, cosine, and tangent. It defines periodic functions as having repeating patterns, with the period being the length of one full pattern. For sine waves, the period is 2. The amplitude is the maximum vertical deviation from the central axis. Graphs of sine, cosine, and tangent are examined over various domains and their characteristics such as zeroes and asymptotes are outlined.
This document discusses converting between degrees and radians for angles and locating angles in the x-y plane based on positive rotation. It provides examples of multiples of common angles like 30, 45, 60, 90 degrees and their radian equivalents, and asks the reader to express answers for questions in a given radian range with positive rotation.
The document provides instructions for solving word problems involving radians. It includes 4 problems: 1) finding the central angle given an arc length of 1.8 cm and radius of 1.2 cm, 2) finding the central angle given the arc length is 1/3 the circumference, 3) calculating the radius given an arc length of 15 cm and central angle of 142 degrees, 4) finding the central angle given an arc length of 14 inches and a frisbee diameter of 12 inches.
The document discusses solving trigonometric equations for exact values over specified intervals. It provides examples of finding trig function values given relationships between other trig functions, such as finding the angle θ if cosine is a certain value and tangent is positive. It also discusses finding the general solution, or all possible values, for trigonometric equations over the real number domain.
This document discusses graphing composite functions. It provides examples of composing two functions f(x) and g(x), such as finding (f ◦ g)(x) and (g ◦ f)(x), and graphing the resulting composite functions. The document emphasizes determining the domains of the composite functions by considering the restrictions on the domains of the original functions. It also gives examples of finding equations for composite functions f(g(x)) and g(f(x)) and stating their domains and ranges.
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.
This document describes how to transform the graph of a function y=f(x) by applying a combination of vertical and horizontal stretches/compressions and shifts. Specifically, it states that applying the transformation h(x)=af(bx-c)+d results in graphically shifting the original function horizontally by c and vertically shifting and stretching/compressing by a, b, and d.
This document discusses trigonometric ratios and the unit circle. It introduces the CAST rule which states that for any point P(θ) on the unit circle, the x-coordinate is equal to cosθ and the y-coordinate is equal to sinθ. It also defines the reciprocal trig functions secant, cosecant, and cotangent in terms of sine, cosine, and tangent. Examples are provided for finding points on the unit circle given trigonometric ratio values.
Functions can be combined by graphing them on the same coordinate plane and looking at the result. When adding functions, the values of the y-coordinates are summed at each x-value. When multiplying functions, the values are multiplied correspondingly. Combining functions through graphing allows us to understand how the inputs are transformed through various compositions of operations.
This document discusses how to find the sum and differences of functions. The sum of two functions can be found by adding the y-coordinates of each function. For example, if f(x) = 2x + 3 and g(x) = x^2 - x - 5, then h(x) = (f + g)x is found by adding the y-coordinates of f(x) and g(x). The difference of two functions is found by subtracting the y-coordinates.
The document discusses the unit circle and circular functions. It defines the unit circle as having a radius of 1 unit and describes coterminal angles as angles that share the same terminal side or arm. It then explains there are three systems for measuring angles: degrees, radians, and gradians, and provides formulas for converting between degrees and radians.
Composite functions refer to combining two functions where the output of one acts as the input of the other. The notation used is (f ◦ g)(x) which reads "f composed with g of x", where the inner function g(x) is evaluated first, then substituted into the outer function f(x). Examples show how to evaluate composite functions by first substituting the inner function and then evaluating the outer function.
El documento describe el sentido del gusto. Explica que los quimiorreceptores de la lengua detectan sustancias químicas solubles y transmiten esta información al cerebro para que perciba diferentes sabores. Describe las papilas gustativas y botones gustativos que se encuentran en la lengua y otras áreas de la boca y faringe, los cuales contienen los receptores del gusto. También explica que diferentes zonas de la lengua son más sensibles a ciertos sabores como dulce, salado, ácido y amargo.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms for those who already suffer from conditions like anxiety and depression.
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.
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.
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.
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.
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!.
This document discusses trigonometric functions such as sine, cosine, and tangent. It defines periodic functions as having repeating patterns, with the period being the length of one full pattern. For sine waves, the period is 2. The amplitude is the maximum vertical deviation from the central axis. Graphs of sine, cosine, and tangent are examined over various domains and their characteristics such as zeroes and asymptotes are outlined.
This document discusses converting between degrees and radians for angles and locating angles in the x-y plane based on positive rotation. It provides examples of multiples of common angles like 30, 45, 60, 90 degrees and their radian equivalents, and asks the reader to express answers for questions in a given radian range with positive rotation.
The document provides instructions for solving word problems involving radians. It includes 4 problems: 1) finding the central angle given an arc length of 1.8 cm and radius of 1.2 cm, 2) finding the central angle given the arc length is 1/3 the circumference, 3) calculating the radius given an arc length of 15 cm and central angle of 142 degrees, 4) finding the central angle given an arc length of 14 inches and a frisbee diameter of 12 inches.
The document discusses solving trigonometric equations for exact values over specified intervals. It provides examples of finding trig function values given relationships between other trig functions, such as finding the angle θ if cosine is a certain value and tangent is positive. It also discusses finding the general solution, or all possible values, for trigonometric equations over the real number domain.
This document discusses graphing composite functions. It provides examples of composing two functions f(x) and g(x), such as finding (f ◦ g)(x) and (g ◦ f)(x), and graphing the resulting composite functions. The document emphasizes determining the domains of the composite functions by considering the restrictions on the domains of the original functions. It also gives examples of finding equations for composite functions f(g(x)) and g(f(x)) and stating their domains and ranges.
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.
This document describes how to transform the graph of a function y=f(x) by applying a combination of vertical and horizontal stretches/compressions and shifts. Specifically, it states that applying the transformation h(x)=af(bx-c)+d results in graphically shifting the original function horizontally by c and vertically shifting and stretching/compressing by a, b, and d.
This document discusses trigonometric ratios and the unit circle. It introduces the CAST rule which states that for any point P(θ) on the unit circle, the x-coordinate is equal to cosθ and the y-coordinate is equal to sinθ. It also defines the reciprocal trig functions secant, cosecant, and cotangent in terms of sine, cosine, and tangent. Examples are provided for finding points on the unit circle given trigonometric ratio values.
Functions can be combined by graphing them on the same coordinate plane and looking at the result. When adding functions, the values of the y-coordinates are summed at each x-value. When multiplying functions, the values are multiplied correspondingly. Combining functions through graphing allows us to understand how the inputs are transformed through various compositions of operations.
This document discusses how to find the sum and differences of functions. The sum of two functions can be found by adding the y-coordinates of each function. For example, if f(x) = 2x + 3 and g(x) = x^2 - x - 5, then h(x) = (f + g)x is found by adding the y-coordinates of f(x) and g(x). The difference of two functions is found by subtracting the y-coordinates.
The document discusses the unit circle and circular functions. It defines the unit circle as having a radius of 1 unit and describes coterminal angles as angles that share the same terminal side or arm. It then explains there are three systems for measuring angles: degrees, radians, and gradians, and provides formulas for converting between degrees and radians.
Composite functions refer to combining two functions where the output of one acts as the input of the other. The notation used is (f ◦ g)(x) which reads "f composed with g of x", where the inner function g(x) is evaluated first, then substituted into the outer function f(x). Examples show how to evaluate composite functions by first substituting the inner function and then evaluating the outer function.
El documento describe el sentido del gusto. Explica que los quimiorreceptores de la lengua detectan sustancias químicas solubles y transmiten esta información al cerebro para que perciba diferentes sabores. Describe las papilas gustativas y botones gustativos que se encuentran en la lengua y otras áreas de la boca y faringe, los cuales contienen los receptores del gusto. También explica que diferentes zonas de la lengua son más sensibles a ciertos sabores como dulce, salado, ácido y amargo.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms for those who already suffer from conditions like anxiety and depression.
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.
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.
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.
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.
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!.
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.
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.
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.
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.
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.
This document discusses combining functions by graphing. When two functions f(x) and g(x) are combined, their graphs are overlayed on the same coordinate plane. The result is a new combined function where the output is determined by applying both functions f(x) and g(x) to the same input x.
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.
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.
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.
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.
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.