Generally, functions in the form of y=ax2+bx+c(a, b and c are constants, and a≠0) are called quadratic functions of X. For example, y=3x2, y=3x2-2, y=2x2+x- 1 are all quadratic functions.
Note: (1) quadratic function is a quadratic form about independent variables, quadratic coefficient A must be a non-zero real number, that is, a≠0, while B and C are arbitrary real numbers, and the expression of quadratic function is algebraic expression;
(2) the quadratic function y=ax2+bx+c(a, b, c are constants, a≠0), and the range of the independent variable x is all real numbers;
(3) When b=c=0, the quadratic function y=ax2 is the simplest quadratic function;
(4) Whether a function is a quadratic function can only be concluded by comparing it with the simplified definition. For example, y=x2-x(x- 1) becomes y=x after simplification, so it is not a quadratic function.
Images and Properties of Quadratic Function y=ax2
The image of (1) function y=ax2 is a curve symmetrical about y axis, which is called parabola. In fact, all images of quadratic function are parabolas.
The image of quadratic function y=ax2 is a parabola, which is symmetrical about Y axis, and the vertex coordinate is (0,0).
① when a >; 0, the parabola y=ax2 has an upward opening, and the curve descends from left to right on the left side of the symmetry axis; On the right side of the symmetry axis, the curve rises from left to right, and the vertex is the lowest point on the parabola, that is, when a >; 0, the function y=ax2 has the following properties: when x0, the function y increases with the increase of x; When x=0, the function y=ax2 takes the minimum value, and the minimum value y = 0；;
② when a
③ When |a| is larger, the opening of parabola is smaller, and when |a| is smaller, the opening of parabola is larger.
(2) Determine the expression of quadratic function y=ax2.
Because the quadratic function y=ax2 contains only one coefficient a, the value of a can be obtained only by giving a pair of corresponding values of x and y. 。
Number of intersections between parabola and x axis
δ= b^2-4ac>； 0, parabola and x axis have two intersections.
When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.
δ= b^2-4ac<； 0, the parabola has no intersection with the x axis.
Important knowledge points of mathematical algebraic expressions
1. Algebraic expression: Algebraic expression is a general term for monomials and polynomials.
2. Algebraic expression addition and subtraction
Algebraic expression addition and subtraction operation, if you encounter parentheses, first remove the parentheses, and then merge similar items.
(1) bracket removal: add and subtract several algebraic expressions. If there are brackets, remove them first, and then merge similar items.
If the factor outside the brackets is positive, the symbols in the original brackets are the same after the brackets are removed.
If the factor outside the bracket is negative, the sign in the original bracket is opposite after the bracket is removed.
(2) Merge similar items:
After merging similar items, the coefficients of the obtained items are the sum of the coefficients before merging, and the letter part remains unchanged.
3. Monomial: An algebraic expression composed of the product of numbers or letters is called a monomial, and a single number or letter is also called a monomial.
4. Polynomials: Algebraic expressions composed of several monomials are called polynomials.
5. A power with the same radix refers to a power with the same radix.
6. Power with the same base: power with the same base, the base is unchanged, and the index is added.
7. Power Law: Power, constant cardinal number, exponential multiplication.
8. Power of product: the power of product. First, multiply each factor in the product separately, and then multiply the obtained power.
9. Multiply the monomial with the monomial
Multiply the monomial with the monomial, and multiply them by their coefficients and the same base respectively. For letters contained only in the monomial, they are used as a factor of the product together with its index.
10. Multiplication of monomial and polynomial
Multiplying a polynomial by a monomial is to multiply each term of a polynomial by a monomial, and then add the products.
1 1. Polynomial times polynomial
Multiply polynomials by multiplying each term of one polynomial by each term of another polynomial, and then add the products.
12. Division with the same base: division with the same base, constant base and exponential subtraction.
13. The monomial is divided by the monomial: the monomial is divided by the coefficient and same base powers respectively as the factor of the quotient; For the letter only contained in the division formula, it is used as the factor of quotient together with its index.
14. Polynomial divided by monomial: Polynomial divided by monomial, first divide each term of polynomial by this monomial, and then add the obtained quotients.
Special trigonometric function value of junior high school mathematics
1.cos30 = = root number 3/2.
2.sin260 +cos260 = 1。
3.2sin30 +tan45 =2。
4.tan45 = 1。
5.cos60 +sin30 = 1。