Chapter 2 Algebra and Functions Review代数与函数

1、Chapter 2 Algebra and functions review,1.Operations on algebraic expressions,You will need be able to apply the basic operations of arithmetic-addition,subtraction,multiplication and division-to algebraic expression.,For example :,2.Factoring,The types of factoring included on the mathematics sectio

2、n are:,Difference of two squares:,Finding common factors,as in:,Factoring quadratics:,3.Exponents,Definition,Three Points to Remember,1.When multiplying expressions with the same base,add the exponents:,This rule also holds for exponents that are not positive integers:,2.When dividing expressions wi

3、th the same base,subtract exponents:,3.When a number raised to an exponent is raised to a second exponent, multiply the exponents:,4.Evaluating expressions with exponents and roots,You will need to know how to evaluate expressions involving exponents and roots.,For instance:,5.Solving equations,Most

4、 of the equations that you will need to solve are linear equations. Equations that are not linear can usually be solved by factoring or by inspection.,1)Working with “ Unsolvable” Equations,2) Solving for one variable in terms of another,3) Solving equations involving radical expressions,6.Absolute

5、value,The absolute value of a number is its distance from zero on the number line. The absolute value of the number is denoted,You may be asked to work with expressions and solve equations that involve absolute value.,For example :,7.Direct translation into mathematical expressions,Many words requir

6、e you to translate the verbal description of a mathematical fact or relationship into mathematical terms.,Always read the word problem carefully and double-check that you have translated it exactly. For example:,8.Inequalities,An inequality is a statement that one quantity is greater than or less th

7、an another. Inequalities are shown using four symbols:,9.Systems of linear equations and inequalities,You may be asked to solve systems of two or more linear equations or inequalities. For example:For what values of a and b are the following equations both true?,Substitution and Elimination,10.Solvi

8、ng quadratic equations by factoring,You may be asked to solve quadratic equations that can be factored.(You will not be expect to know the quadratic formula),For instance:For what values of,11.Rational equations and inequalities,A rational algebraic expression is the quotient of two expressions.An e

9、xample is,You may be asked to solve equations or inequalities involving such expressions.For example :For what value of x is the following equation true?,12.Direct and inverse variation,The quantities x an d y are directly proportional if for some constant k.,For example: x and y are directly propor

10、tional.When the value x is 10, y is equal to -5. If x=3, what is y?,The quantities x an d y are inversely proportional if for some constant k.,For example:If xy=4 ,show that x and y are inversly proportional.,Example 1 The number of juice packs that Rambling Pines Day Camp buys each day is directly

11、proportional to the number of children at the camp that day.If the camp bought 150 juice boxes yesterday for 60 children,how many juice boxes will they buy today for 52 children?,Example 2 If c and d are inversely proportional and c=10 when d=6, what is d when c=30?,13.Word problems,Example1 The pri

12、ce of a sweater went up 20% since last year. If last years price was x, what is this years price in terms of x?,Example 2 One year ago,an average restaurant meal cost 12.Today,the average restaurant meal cost 15.By what percent has the cost of the average restaurant meal increased?,Example 3 The ave

13、rage height(arithmetic mean) of 4 members of a 6-person volleyball team is 175cm.What does the average height in centimeters of the other 2 players have to be if the average height of the entire team equals 180cm?,Example 4 A car traveling at an average rate of 55 kilometers oer hour made a trip in

14、6 hours.If it had traveled at an average rate of 50 kilometers per hour, the trip would have taken how many minutes longer?,14. Functions,1)Function notation and evaluation A function can be thought of as a rule or formula that tells how to associate the elements in one set (the domain) with the ele

15、ments in another set (the range).,For example, the squaring function can be thought of as the rule take the square of x or as the rule .,The squaring function can be writtrn in function notation as,2)Domain and range,The domain of a function is the set of all the values for which the function is def

16、ined. The range of a function is the set of all the values that are the output, or results, of applying the function.,Example What are the domain and range of,3)Using new definitions,For functions, especially those involving more than one variable, a special symbol is sometimes introduced and define

17、d. These symbols generally have unusual looking signs,so you wont confuse them with standard mathematical symbols.,You may be asked to compare two values,each of which requires the uses of the symbol. You may be asked the evaluate an expression than involves multiplying ,dividing, adding,squaring or

18、 subtracting terms that involve the symbol. You may be asked to solve an equation that involves the use of the symbol.,Some questions will ask you to apply the definition of the symbol to morecomplicated situation.For instance:,4)Functions as models,Functions can be used as models of real-life situa

19、tion.Here is an example:,The temperature in City X is W(t) degrees Fahrenheit t hours after sundown at 5:00 p.m. The function W(t) is given by,You can draw a graph of this function:,5)Linear functions: their equations and graphs,You may be asked to answer questions onvolving linear equations and the

20、ir graphs.Therefore, you will need to understand the concepts of slope and intercepts.,is a linear function,and the graph of in the xy-plane is a line with slope m and y-intercept b.,In the figure above,points A,B and C lie on a vertical line segment that intersects line at B and line p at C.OA=1,OE

21、=1,BA=1 and BC=2.What are the coordinates of the point D where lines and p intersect?,For example,6)Quadratic functions: their equations and graphs,You may be asked to answer questions involving quadratic equations and their graphs.You will need be able to identify some of the basic features of the graph of a quadratic equation,such as its highest or lowest point,its zeros and its direction.,For example ,which of the following could be the graph of,7)Qualitative behavior of graphs and func