Chapter1.1
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The document provides examples and explanations for evaluating algebraic expressions. It introduces key vocabulary like expression, variable, numerical expression, algebraic expression, and evaluate. Examples are provided for evaluating expressions with one variable, two variables, and applications involving converting between Celsius and Fahrenheit temperatures. Practice problems are included at the end to evaluate expressions for given variable values.
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This document provides examples and explanations for adding and subtracting integers. It discusses subtracting a positive or negative integer by adding the opposite integer. Additional examples apply integer operations to real-world contexts like architecture and aviation. Practice problems at the end ask students to subtract integers and evaluate expressions with integer variables.
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The document discusses solving linear equations by using addition, subtraction, multiplication, and division properties. It provides examples of solving single-step and multi-step equations, including equations with decimals that require rounding. It also covers rewriting formulas to make one variable a function of another and solving real-world application problems involving formulas.
Chapter3.2
This document provides examples and explanations for combining like terms in algebraic expressions. It includes examples of combining like terms with one variable, two variables, and using the distributive property to simplify expressions. The examples are followed by a short quiz with 5 problems involving combining like terms and simplifying expressions.
Variable and Algebraic Expressions
1. The document provides examples and explanations for evaluating algebraic expressions by substituting values for variables.
2. It gives examples of evaluating expressions involving addition, subtraction, multiplication, division, and order of operations.
3. Students are asked to evaluate expressions for given variable values to check their understanding.
Math 116 pres. 1
CHAPTER 2 Whole Numbers: Introducing a Mathematical System
2.3 Algebraic Extensions of Order of Operations to Polynomials
Module For Mathematics
The document provides information about algebraic expressions and polynomials. It begins by explaining that algebra uses a language of numbers, variables, and symbols to represent quantitative relationships. The document then covers topics like equivalent expressions, simplifying expressions, evaluating expressions, and different types of polynomials. It aims to help students acquire skills in performing operations with polynomials and using those skills to simplify, evaluate, and solve algebraic expressions and problems.
Fst ch3 notes
The document discusses various mathematical concepts related to functions and graphs including:
1) Transformations of graphs such as translations, reflections, and rotations. It also discusses parent functions and their derivatives.
2) Examples of graphing functions after applying transformations to translate, scale, or reflect the original graphs. Equations are provided for the transformed graphs.
3) Theorems related to how statistics of data change after translations or scale changes. For example, the mean, median and mode change proportionally but variance, standard deviation, and range change in specific ways.
4) Concepts involving inverse functions, including using the horizontal line test to determine if an inverse is a function and notations for inverse functions
Successful Minds,Making Mathematics number patterns &sequences Simple.
This document provides information about number patterns and sequences. It discusses arithmetic sequences where the difference between consecutive terms is constant, and quadratic patterns where the second differences are constant. Examples are provided to demonstrate how to determine the general term of a sequence from a given pattern. Practice problems with solutions allow the reader to apply the concepts to find missing terms, general terms, and values of specific terms.
Ebook 1
This document provides information about number patterns and sequences. It discusses arithmetic sequences where the difference between consecutive terms is constant, and quadratic patterns where the second differences are constant. Examples are provided to demonstrate how to determine the general term of a sequence from a given pattern. Practice problems with solutions allow the reader to apply the concepts to find missing terms, general terms, and values of specific terms.
CollegeAlgebra_02_PolynomialandRational_.pptx
Polynomial and rational expressions can be added, subtracted, multiplied, divided, and factored using specific steps and properties. A polynomial is an expression involving variables and coefficients that can be written in standard form. Polynomials can be added and subtracted by combining like terms. Multiplication uses the distributive property or FOIL method for binomials. Rational expressions are quotients of polynomials that can be simplified by cancelling common factors. Specific factoring techniques like GCF, difference of squares, and grouping are used to factor polynomial expressions into their prime factors.
Algebra 2 Section 4-4
This document discusses polynomial functions. It defines key terms like polynomial in one variable, leading coefficient, and polynomial function. It provides examples of power functions of varying degrees like quadratic, cubic, quartic and quintic functions. The document also includes examples of evaluating polynomial functions, finding degrees and leading coefficients, graphing polynomial functions from tables of values, and describing properties of graphs.
THE BINOMIAL THEOREM
THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)
Ppt 1.1 and 1.2
The document provides examples and exercises on evaluating and simplifying algebraic expressions involving real numbers. It begins with examples of evaluating powers and algebraic expressions by substituting values for variables. Following examples show using verbal models to write and evaluate algebraic expressions representing word problems. The document also provides examples and guided practice on simplifying expressions by combining like terms. One example gives a word problem about the cost of printing digital photos, writing an expression for the total cost, and evaluating it for a given value of n.
Chapter2.3
The document provides examples and explanations for performing arithmetic operations with rational numbers including integers, fractions, and decimals. It demonstrates how to add, subtract, multiply, and divide rational numbers, as well as how to evaluate expressions involving rational numbers. The examples are aligned with California state math standards for performing operations with rational numbers.
Chapter1.2
This document provides examples for translating between word phrases, algebraic expressions, and word problems. It shows how to write an algebraic expression from a word phrase or vice versa. Examples are given for writing an expression or word problem to match a given expression and evaluating expressions for given variable values in word problems. The lesson quiz asks students to write expressions from word phrases and an expression and word problem to evaluate a given expression.
390 Guided Projects Guided Project 31 Cooling cof.docx
This document describes Euler's method for numerically approximating solutions to differential equations. Euler's method works by breaking the time interval into discrete steps and using the slope of the direction field at each point to estimate the solution at the next time step. The document provides an example of using Euler's method to solve the initial value problem y'(t) = -2y + 4, y(0) = 1, with time step Δt = 0.2 and number of steps N = 5. It includes a table to fill in with the computed approximations to y(t) at each time step.
390 Guided Projects Guided Project 31 Cooling cof.docx
390 Guided Projects
Guided Project 31: Cooling coffee
Topics and skills: Derivatives, exponential functions
Imagine pouring a cup of hot coffee and letting it cool at room temperature. How does the temperature of the
coffee decrease in time? How long must you wait until the coffee is cool enough to drink? When should you
add an ounce of cold milk to the coffee to accelerate the cooling as much as possible?
A fairly accurate model to describe the temperature changes in a conducting object is Newton’s Law of
Cooling. Suppose that at time t ≥ 0 an object has a temperature of T(t). The Law of Cooling says that the rate at
which the temperature of the object increases or decreases is given by
( ( ) ) , (1)
dT
k T t A
dt
= − −
where A is the ambient (surrounding) temperature and k > 0 is a constant called the conductivity (which is a
property of the cooling object). Newton’s Law of Cooling assumes that the cooling body has a uniform
temperature throughout its interior. This is not strictly accurate, as a cooling body loses heat through its surface.
1. Explain in words what equation (1) means. Specifically, in terms of T and A, when is 0
dT
dt
> and when is
0
dT
dt
< ? For the case of hot coffee cooling to room temperature, which case do you expect to see?
2. Verify by substitution that the solution to equation (1) subject to the initial condition T(0) = T0 is
0( ) ( ) . (2)
ktT t A T A e−= + −
3. Before graphing the temperature function, use equation (2) to evaluate T(0) and limt→∞ T(t). Are these the
values you expect?
4. Consider the case of a cup of hot coffee cooling with an ambient room temperature of A = 60◦ F and the
initial temperature of the coffee is T0 = 200
◦ F. Use a graphing utility to plot the temperature function for
k = 0.3, 0.2, 0.1, and 0.05. Comment on how the curves change with k. Do larger values of k produce faster
or slower rates of temperature change?
5. For the values of A and T0 in Step 4, estimate the value of k that describes the case in which the coffee
cools to 100 degrees in 10 minutes.
Here is an interesting question. Suppose you want to cool your hot coffee to 100◦ F as quickly as possible.
Suppose also that you have one ounce of cold milk with a temperature of 40◦ F that you can add to the
cooling coffee at any time. When should you add the milk to cool the coffee to 100◦ F as quickly as
possible?
6. We need to make an assumption about the effect of cold milk on the temperature of the coffee. A
reasonable assumption is that when milk is added to coffee, the temperature of the coffee immediately
decreases to the average of the coffee temperature and the milk temperature, where the average is weighted
by the volumes. So if we add 1 ounce of milk with temperature Tm to 8 ounces of coffee with temperature
T, the temperature of the mixture will be
1 8 8
. (3)
1 8 9
m m
new
T T T T
T
⋅ .
Linear equations inequalities and applications
This document provides information about chapter 2 of a math textbook. It covers linear equations, formulas, and applications. Section 2-1 discusses solving linear equations, including using properties of equality and identifying conditional, identity, and contradictory equations. Section 2-2 introduces formulas and how to solve them for a specified variable. Section 2-3 explains how to translate words to mathematical expressions and equations, and how to solve applied problems using a six step process. An example at the end solves a word problem about baseball players' home run totals.
Carmen sanchez
En esta ocasión te presentamos esta información sobre expresiones algebraicas, además de ello ejercicios variados que te pueden ser mucha utilidad
Sequence and Series
FREE MCA NATIONAL LEVEL MOCK TEST
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Lecture 3 (algebraic expressions)
This document provides information and examples about algebraic expressions and evaluating expressions using order of operations. It includes:
- Definitions of variables, expressions, operations like addition and multiplication.
- Examples of writing expressions for word phrases and evaluating numerical expressions using order of operations rules like BODMAS/PEMDAS.
- Questions asking to write expressions from word phrases, evaluate expressions for given variable values, and evaluating variable expressions.
The document covers basic topics of writing and evaluating algebraic expressions and the order in which operations should be performed. It provides examples to illustrate definitions and multiple practice questions.
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Successful Minds,Making Mathematics number patterns &sequences Simple.
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3 3 i can add and subtract decimals
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New week 1
The document provides information about a pre-algebra class, including:
1) Instructions for students to write down their homework, review class expectations, and get out notebooks.
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The document contains instructions and information about using similar triangles to estimate heights and distances that are difficult to measure directly. It discusses using shadows and mirrors to form similar triangles and estimate heights of objects. Measurements are taken of shadows, mirrors, and distances that allow heights to be calculated through properties of similar triangles. Students work through examples estimating heights of buildings using these methods.
11.6
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2) Solving a rational equation with rational expressions on both sides. The solution appears to be 2, but it is an extraneous solution.
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11.4
1) To add or subtract rational expressions with the same or different denominators, we first find the least common denominator (LCD), which is the least common multiple of the denominators.
2) We then rewrite the rational expressions with the LCD as the common denominator before adding or subtracting the numerators.
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Chapter1.1
- 1. Warm Up California Standards Lesson Presentation Preview
- 2. Warm Up Simplify. 1. 21 – 2(3) 15 31 21 20.4 15.3 5. 7(2.9) – 5 4. 6(1.4) + 12 3. 2(9) + (3) 2. 4 + 3 9
- 3. AF1.2 Use the correct order of operations to evaluate algebraic expressions such as 3(2 x + 5) 2 . AF1.4 Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality , expression, constant ) correctly. California Standards
- 4. Vocabulary expression variable numerical expression algebraic expression evaluate
- 5. An expression is a mathematical phrase that contains operations, numbers, and/or variables . A variable is a letter that represents a value that can change or vary. There are two types of expressions: numerical and algebraic . A numerical expression does not contain variables. An algebraic expression contains one or more variables. x + 2 4 n p – r 3 + 2 4(5) 27 – 18 Algebraic Expressions Numerical Expressions
- 6. To evaluate an algebraic expression, substitute a given number for the variable. Then use the order of operations to find the value of the resulting numerical expression.
- 7. Evaluate the expression for the given value of the variable. Additional Example 1A: Evaluating Algebraic Expressions with One Variable Substitute 12 for x. Subtract. x – 5 for x = 12 12 – 5 7
- 8. Order of Operations PEMDAS: 1. P arentheses 2. E xponents 3. M ultiply and D ivide from left to right. 4. A dd and S ubtract from left to right. Remember!
- 9. Evaluate the expression for the given value of the variable. Additional Example 1B: Evaluating Algebraic Expressions with One Variable Substitute 4 for y. Multiply. 2 y + 1 for y = 4 2( 4 ) + 1 8 + 1 Add. 9
- 10. Evaluate the expression for the given values of the variable. Additional Example 1C: Evaluating Algebraic Expressions with One Variable 6( n + 2) – 4 for n = 5, 6, 7 6( 5 + 2) – 4 6(7) – 4 42 – 4 38 6( 6 + 2) – 4 6(8) – 4 48 – 4 44 6( 7 + 2) – 4 6(9) – 4 54 – 4 50 7 6 5 Subtract Multiply Parentheses Substitute n
- 11. Evaluate each expression for the given values of the variable. Check It Out! Example 1 Substitute 3 for x. Add. A. x + 5 for x = 3 3 + 5 8 Substitute 11 for c. Multiply. B. 4 c + 1 for c = 11 4( 11 ) + 1 44 + 1 Add. 45
- 12. Evaluate the expression for the given values of the variable. Check It Out! Example 1C 5( n + 3) – 2 for n = 3, 4, 5 5( 3 + 3) – 2 5(6) – 2 30 – 2 28 5( 4 + 3) – 2 5(7) – 2 35 – 2 33 5( 5 + 3) – 2 5(8) – 2 40 – 2 38 5 4 3 Subtract Multiply Parentheses Substitute n
- 13. Evaluate each expression for the given values of the variables. Additional Example 2: Evaluating Algebraic Expressions with Two Variables Substitute 2 for x and 1 for y. Multiply. A. 4 x + 3 y for x = 2 and y = 1 4( 2 ) + 3( 1 ) 8 + 3 Substitute 3 for r and 5 for p. Multiply. B. 9 r – 2 p for r = 3 and p = 5 9( 3 ) – 2( 5 ) 27 – 10 Add. 11 Subtract. 17
- 14. Evaluate each expression for the given values of the variables. Check It Out! Example 2 Substitute 3 for x and 4 for y. Multiply. A. 3 x + 2 y for x = 3 and y = 4 3( 3 ) + 2( 4 ) 9 + 8 Substitute 2 for q and 2 for r. Multiply. B. 8 q – 3 r for q = 2 and r = 2 8( 2 ) – 3( 2 ) 16 – 6 Add. 17 Subtract. 10
- 15. Use the expression 1.8 c + 32 to convert the temperature from degrees Celsius to degrees Fahrenheit. Additional Example 3A: Physical Science Application Substitute 100 for c. Multiply. Boiling point of water at sea level: 100°C 1.8( 100 ) + 32 180 + 32 1.8c + 32 Add. 212 100 ° C = 212 ° F Water boils at 212 ° F at sea level.
- 16. Additional Example 3B: Physical Science Application Substitute 85 for c. Multiply. Boiling point of water at an altitude of 4400 t meters: 85°C 1.8( 85 ) + 32 153 + 32 1.8c + 32 Add. 185 85 ° C = 185 ° F Water boils at 185 ° F at 4400 m. Use the expression 1.8 c + 32 to convert the temperature from degrees Celsius to degrees Fahrenheit.
- 17. Convert the temperature from degrees Celsius to degrees Fahrenheit. Check It Out! Example 3A Substitute 50 for c. Multiply. Europe’s highest recorded temperature (Seville, Spain): 50°C 1.8( 50 ) + 32 90 + 32 1.8c + 32 Add. 122 50 ° C = 122 ° F The highest recorded temperature in Europe is 122 ° F.
- 18. Use the expression 1.8 c + 32 to convert the boiling point temperature from degrees Celsius to degrees Fahrenheit. Check It Out! Example 3B Substitute 10 for c. Multiply. 10°C 1.8( 10 ) + 32 18 + 32 1.8c + 32 Add. 50 10 ° C = 50 ° F
- 19. Lesson Quiz Evaluate each expression for the given value(s) of the variables. 1. 6 x + 9 for x = 3 2. x + 14 for x = 8 3. 4 x + 3 y for x = 2, y = 3 4. y (27 – x ) for x = 19, y = 6 5. If n is the amount of money in a savings account, then the expression n + 0.03 n can be used to find the amount in the account after it has earned interest for one year. Find the total in the account after one year if $500 is the initial amount. 27 22 17 48 $515