sat的知识点总结

Algebra

1. Concept

Powers of Numbers

Roots of numbers

2. Powers

3. Exponents, roots, and real number line.

Raising numbers to powers can have surprising effects on the size or sign of the base number. This is one of the test-makers` favourite areas! The impact of raising a number to an exponent depends on the region on the number line where the number and exponent fall. Here are the four regions you need to consider;

(1) Less than -1

(2) Between -1 and 0

(3) Between 0 and 1

(4) Greater than 1

As with exponents, the root of a number can bear a surprising relationship to the size and sign of the number. Here are our observations you should remember.

(1) If n>1 , (the higher the roots, the higher the value.)

(2) The square root of any negative number is an imaginary number, not a real number. Remember: you won`t encounter any imaginary numbers on the SAT.

(3) Every positive number has two square roots: a negative number and a positive number (with the same absolute value). The same holds true for all other even-numbered roots of positive numbers.

(4) Every negative number has exactly one cube root, and that root is a negative number. The same holds true for all other odd-numbered roots of negative numbers.

(5) Every positive number has only one cube root, and that root is always a positive number. The same holds true for all other odd-numbered root of positive numbers.

4. The Operation Rule of Radicals

Combining Radicals

(1) Addition and Subtraction: if a term under a radical is being added to or subtracted from a term under a different radical, you cannot combine the two terms under the same radical.

(2) Multiplication and Division: terms under different radicals can be combined under a common radical if one term is multiplied or divided by the other, but only if the root is the same.

Simplifying Radicals

On the SAT, always look for the possibility of simplifying radicals by moving part of what`s inside the radical to the outside. Check inside your square-root radicals for factors that are squares of nice tidy numbers (especially integers).

5. Function

(1) Liner Equation with One Variable

Algebraic expressions are exactly used to form equations, which set two expressions equal to each other. Most equations you`ll see on the SAT are linear equations, in which the variables don`t come with exponents. To sole any liner equation containing one variable, your goal is always the same; the unknown (variable) on one size of the equation. To accomplish this, you may need to perform one or more of the following operations on both sides, depending on the equation.

Add or subtract the same term both sides of the equation

Multiply of divided both sides of the equation by the same non-zero term.

If each side of the equation is a fraction, your best bet is to cross-multiply.

Square both sides of the equation to eliminate radical signs.

(2) Linear Equations with Two Variables

The substitution method

The addition-subtraction method

(3) Quadratic equation with one variable

An equation is quadratic if you can express it in this general form: .

In this general form, x is the variable, a, b and c are constants (numbers), , b and c can equal 0. Every quadratic equation has exactly two solutions. (These two solutions are called roots.) All quadratic equations on the SAT can be solved by factoring.

Factorable quadratic equations with one variable

To solve any factorable quadratic equation with one variable, follow these three steps:

(1) put the equation into the standard form: .

(2) Factor the terms on the left side of the equation into two linear expression (with no exponents).

(3) Set each linear expression (root) equal to zero solve for the variable in each one.

6 Series

7 the Motion of the Graph of Function

Functions with similar equations tend to have similar shapes. For instance, functions of the quadratic form have graphs that look like parabolas. You also should know how specific changes to the function equation produce specific changes to the graph. Learn how to recognize basic transformations of functions: shifts and reflections.

(1) Horizontal motion: means that the graph moves left k units while the graph moves right k units.

(2) Vertical motion: means means that the graph moves upward k units while means the graph moves right k units.

(3) Reflections: when the point (x,y) is reflected over the y axis, it becomes (-3,4). That is, the x coordinate is negated. When it is reflected over the x axis, it becomes (3,-4). That is, the y coordinate is negated. Likewise., if the graph of is reflected over the x axis, it becomes .