Day 3 examples b u1f13
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1) The x-intercept of the function y=f(x) is 5, so the x-intercept of y=f(x)^2 is also 5. 2) The coordinates of the point on the graph of y=g(x) are (-3,5), so the coordinates of the point on the graph of y=g(-x) are (3,5). 3) To write the equation of a function f(x) that is shifted down 4 units and compressed vertically by a factor of 2, the equation would be y = 1/2f(x) - 4.
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Day 3 examples b u1f13
1) The x-intercept of the function y=f(x) is 5, so the x-intercept of y=f(-x) is -5.
2) The coordinates of the point on the graph of y=g(x) are (-3,5), so the coordinates of the point on y=g(-x) are (3,5).
3) The equation y=-f(x-4) describes a function f(x) that is shifted down 4 units and compressed vertically by a factor of 2.
Day 4a examples
1. The x-intercept of the function y=f(x) is 5, so the x-intercept of y=f(x)^2 is also 5.
2. The coordinates of a point on the graph of y=g(x) are (-3,5), so the coordinates of a point on the graph of y=h(-x) are (3,5).
3. The equation y=-f(x-4) describes a function that is shifted down 4 units and compressed vertically by a factor of 2 compared to the original function f(x).
Mult div rational exp
Rational expressions can be written as the ratio of two polynomials, where the denominator is not equal to 0. Examples of rational expressions are provided. The document provides examples of simplifying, multiplying, and dividing rational expressions by factoring polynomials in the numerators and denominators. The key steps are to multiply or divide the numerators and denominators separately, then simplify if possible by canceling common factors.
MATEMÁTICA | SEMANA 33 | 3ª SÉRIE | POLINÔMIO DECOMPOSIÇÃO EM FATORES DO 1º ...
This document provides a lesson on factorizing polynomials of the first degree. It defines polynomials and provides examples. It states the fundamental theorem of algebra that any polynomial of degree n can be factored into linear factors. An example problem asks students to identify the factored form of a cubic polynomial given its three roots. The correct answer is identified as (x + 2)(x - 3)(x - 5). Key terms discussed include: term, degree, variable, exponent, and factoring.
Equation of a straight line y b = m(x a)
1. The document discusses finding the equation of a straight line given two points or the gradient and a point.
2. It provides the formula for finding the equation of a line as y - b = m(x - a), where m is the gradient, (a, b) is a point on the line.
3. An example finds the equation of the line connecting points A(2, 6) and B(5, 8) using the formula, getting the equation y = 2/3x + 14/3.
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The document provides examples and explanations of finding the rate of change and linear equations from tables of data points and given slopes and points. It includes examples of using the point-slope formula to write linear equations from a given point and slope, as well as examples of writing linear equations from two given points. Students are prompted to write linear equations for several examples.
Basic Integral
The document discusses integrals and their use in calculating areas and volumes. It defines an integral as combining infinitesimal data to determine displacement, area, volume, and other concepts. The document provides an example of calculating the indefinite integral of the function f(x)=x as 1/2x^2 + C and the definite integral from 0 to 3 of f(x)=x as 4.5 by using the integral formula and verification with the area of a rectangle formula.
Semana 33 matemática 3ª série - polinômio - decomposição em fatores do 1º grau
This document provides a lesson on factorizing polynomials of the first degree. It defines polynomials and gives examples. It states the fundamental theorem of algebra that any polynomial of degree n can be decomposed into linear factors. An example problem is given where a polynomial p(x) of degree 3 has roots of -2, 3, and 5. The correct expression representing this polynomial is identified as p(x) = (x + 2)(x - 3)(x - 5). Key terms like term, degree, variable, and exponent are highlighted for attention.
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Day 3 examples b u1f13
1) The x-intercept of the function y=f(x) is 5, so the x-intercept of y=f(-x) is -5.
2) The coordinates of the point on the graph of y=g(x) are (-3,5), so the coordinates of the point on y=g(-x) are (3,5).
3) The equation y=-f(x-4) describes a function f(x) that is shifted down 4 units and compressed vertically by a factor of 2.
Day 4a examples
1. The x-intercept of the function y=f(x) is 5, so the x-intercept of y=f(x)^2 is also 5.
2. The coordinates of a point on the graph of y=g(x) are (-3,5), so the coordinates of a point on the graph of y=h(-x) are (3,5).
3. The equation y=-f(x-4) describes a function that is shifted down 4 units and compressed vertically by a factor of 2 compared to the original function f(x).
Mult div rational exp
Rational expressions can be written as the ratio of two polynomials, where the denominator is not equal to 0. Examples of rational expressions are provided. The document provides examples of simplifying, multiplying, and dividing rational expressions by factoring polynomials in the numerators and denominators. The key steps are to multiply or divide the numerators and denominators separately, then simplify if possible by canceling common factors.
MATEMÁTICA | SEMANA 33 | 3ª SÉRIE | POLINÔMIO DECOMPOSIÇÃO EM FATORES DO 1º ...
This document provides a lesson on factorizing polynomials of the first degree. It defines polynomials and provides examples. It states the fundamental theorem of algebra that any polynomial of degree n can be factored into linear factors. An example problem asks students to identify the factored form of a cubic polynomial given its three roots. The correct answer is identified as (x + 2)(x - 3)(x - 5). Key terms discussed include: term, degree, variable, exponent, and factoring.
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2. It provides the formula for finding the equation of a line as y - b = m(x - a), where m is the gradient, (a, b) is a point on the line.
3. An example finds the equation of the line connecting points A(2, 6) and B(5, 8) using the formula, getting the equation y = 2/3x + 14/3.
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The document provides examples and explanations of finding the rate of change and linear equations from tables of data points and given slopes and points. It includes examples of using the point-slope formula to write linear equations from a given point and slope, as well as examples of writing linear equations from two given points. Students are prompted to write linear equations for several examples.
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The document discusses integrals and their use in calculating areas and volumes. It defines an integral as combining infinitesimal data to determine displacement, area, volume, and other concepts. The document provides an example of calculating the indefinite integral of the function f(x)=x as 1/2x^2 + C and the definite integral from 0 to 3 of f(x)=x as 4.5 by using the integral formula and verification with the area of a rectangle formula.
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This document provides a lesson on factorizing polynomials of the first degree. It defines polynomials and gives examples. It states the fundamental theorem of algebra that any polynomial of degree n can be decomposed into linear factors. An example problem is given where a polynomial p(x) of degree 3 has roots of -2, 3, and 5. The correct expression representing this polynomial is identified as p(x) = (x + 2)(x - 3)(x - 5). Key terms like term, degree, variable, and exponent are highlighted for attention.
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This document provides an overview of polynomial decomposition into first degree factors. It defines polynomials and provides examples. It then states the theorem that any polynomial of degree n can be decomposed into factors of degree 1. An example problem is given where a third degree polynomial has roots of -2, 3, and 5. The correct expression representing this polynomial is identified as (x + 2)(x - 3)(x - 5) based on applying the polynomial decomposition theorem.
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Yoga, stress & namasmaran dr. shriniwas kashalikar
The document discusses how a lack of yoga, which integrates the left and right parietal cortex, can lead to an inability to see the interconnections between different fields. This disunity can then cause mismanagement of stress, prejudice, injustice and crimes. While intelligence relies on left brain development, right brain development is also needed for balance and to avoid antisocial behavior. The document asks if regular namasmaran, or remembrance of God, can help integrate the left and right brain, improve stress management, and enhance individual and global life.
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Combs perms review pascals
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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- 2. y = f(x) ----> h(x) = af b(x - c) + d (x, y) ----> (x + c, ay + d) af b(x - c) + d vertical stretch/ compress vertically stretch/ compress horizontally horizontal y = 3f 1/2(x - 4) - 5 in words
- 3. m(x) = 1/2 T(-x) - 3 b) f(x) = -j(1/3x + 1)