Summary of knowledge points of mathematical power function
I. Function definition and definition:
Independent variable x and dependent variable y have the following relationship:
y=kx+b
It is said that y is a linear function of x at this time.
In particular, when b=0, y is a proportional function of x.
Namely: y=kx(k is a constant, k≠0)
Second, the properties of linear function:
The change value of 1.y is directly proportional to the corresponding change value of x, and the ratio is k.
That is: y=kx+b(k is any non-zero real number b, take any real number)
2. When x=0, b is the intercept of the function on the y axis.
Iii. Images and properties of linear functions:
1. Practice and graphics: Through the following three steps.
(1) list;
(2) tracking points;
(3) The connection can be the image of a function-a straight line. So the image of a function only needs to know two points and connect them into a straight line. (Usually find the intersection of the function image with the X and Y axes)
2. Property: any point P(x, y) on the (1) linear function satisfies the equation: y = kx+b.
(2) The coordinate of the intersection of the linear function and the Y axis is always (0, b), and the image of the proportional function always intersects the origin of the X axis at (-b/k, 0).
3. Quadrant where K, B and function images are located:
When k>0, the straight line must pass through the first and third quadrants, and Y increases with the increase of X;
When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.
When b>0, the straight line must pass through the first and second quadrants;
When b=0, the straight line passes through the origin.
When b<0, the straight line must pass through three or four quadrants.
Especially, when b=O, the straight line passing through the origin o (0 0,0) represents the image of the proportional function.
At this time, when k>0, the straight line only passes through the first and third quadrants; When k < 0, the straight line only passes through the second and fourth quadrants.
Fourth, determine the expression of a linear function:
Known point A(x 1, y1); B(x2, y2), please determine the expressions of linear functions passing through points A and B. ..
(1) Let the expression (also called analytic expression) of a linear function be y = kx+b.
(2) Because any point P(x, y) on the linear function satisfies the equation y = kx+b, we can list two equations: y 1 = kx 1+b … ① and y2 = kx2+b … ②.
(3) Solve this binary linear equation and get the values of K and B. ..
(4) Finally, the expression of the linear function is obtained.
Power Function of Knowledge Points in Mathematics Volume I
Definition of power function:
A function in the form of y = x a (a is a constant), that is, a function with the base as the independent variable and the exponent as the dependent variable is called a power function.
Domain and Value Domain:
When a is a different numerical value, the different situations of the domain of the power function are as follows: if a is any real number, the domain of the function is all real numbers greater than 0; If a is a negative number, then X must not be 0, but the definition domain of the function must also be determined according to the parity of Q, that is, if Q is even at the same time, then X cannot be less than 0, then the definition domain of the function is all real numbers greater than 0; If q is an odd number at the same time, the domain of the function is all real numbers that are not equal to 0. When x is different, the range of power function is different as follows: when x is greater than 0, the range of function is always a real number greater than 0. When x is less than 0, only when q is odd and the range of the function is non-zero real number. Only when a is a positive number will 0 enter the value range of the function.
Nature:
Rational number with value. As non-zero, it is necessary to discuss their respective characteristics in several cases:
First of all, we know that if a=p/q, q and p are integers, then x (p/q) = the root of q (p power of x), if q is odd, the domain of the function is r, if q is even, the domain of the function is [0, +∞). When the exponent n is a negative integer, let a=-k, then x = 1/(x k), obviously x≠0, and the domain of the function is (-∞, 0)∩(0, +∞). So it can be seen that the limitation of X comes from two points. First, it can be used as a denominator, but it cannot be used as a denominator.
Rule out two possibilities: 0 and negative number, that is, for x>0, then A can be any real number;
The possibility of 0 is ruled out, that is, for X.
The possibility of being negative is ruled out, that is, for all real numbers with x greater than or equal to 0, a cannot be negative.
To sum up, we can draw that when a is different, the different situations of the power function domain are as follows:
If a is any real number, the domain of the function is all real numbers greater than 0;
If a is a negative number, then X must not be 0, but the definition domain of the function must also be determined according to the parity of Q, that is, if Q is even at the same time, then X cannot be less than 0, then the definition domain of the function is all real numbers greater than 0; If q is an odd number at the same time, the domain of the function is all real numbers that are not equal to 0.
When x is greater than 0, the range of the function is always a real number greater than 0.
When x is less than 0, only when q is odd and the range of the function is non-zero real number.
Only when a is a positive number will 0 enter the value range of the function.
Since x is greater than 0, it is meaningful to any value of a, so the following gives the respective situations of power function in the first quadrant.
You can see:
(1) All graphs pass (1, 1).
(2) When a is greater than 0, the power function monotonically increases, while when a is less than 0, the power function monotonically decreases.
(3) When a is greater than 1, the power function graph is concave; When a is less than 1 and greater than 0, the power function graph is convex.
(4) When a is less than 0, the smaller A is, the greater the inclination of the graph is.
(5)a is greater than 0, and the function passes (0,0); A is less than 0, and the function has only (0,0) points.
(6) Obviously the power function is unbounded.
High school mathematics power function knowledge
Monotonicity of 1. Function (Local Property)
(1) incremental function
Let the domain of the function y=f(x) be I, if for any two independent variables x 1 and x2 in the interval d within the domain I, when x 1,
If the values of any two independent variables in the interval d are x 1, x2, and when x 1f(x2), then f(x) is said to be a decreasing function in this interval. The interval d is called monotonically decreasing interval y=f(x).
Note: the monotonicity of a function is a local property of the function;
(2) the characteristics of image
If the function y=f(x) is increasing function or subtraction function in a certain interval, it is said that the function y=f(x) has (strict) monotonicity in this interval, and the image of increasing function rises from left to right, and the image of subtraction function falls from left to right.
(3) The method of judging monotone interval and monotonicity of function.
(1) Definition method:
A. let x 1, x2∈D, x 1.
B. difference f (x1)-f (x2);
C. deformation (usually factorization and formula);
D, number (that is, judging the positive and negative of the difference f(x 1)-f(x2));
E draw a conclusion (point out the monotonicity of the function f(x) in the given interval d).
(b) Image method (looking up and down from the image)
(c) Monotonicity of composite functions
The monotonicity of the compound function f[g(x)] is closely related to the monotonicity of its constituent functions u=g(x) and y=f(u), and its law is "same increase but different decrease".
Note: The monotone interval of a function can only be a subinterval of its domain, and the intervals with the same monotonicity cannot be summed together to write its union.
8. Parity of function (global property)
(1) even function
Generally speaking, f (-x) = f(x) exists for any x in the domain of function f(x), so f (x) is called even function.
(2) odd function
Generally speaking, F (-x) =-f(x) exists for any X in the definition domain of function f(x), so F (x) is called odd function.
(3) Features of images with parity function
The image of even function is symmetrical about y axis; Odd function's image is symmetrical about the origin.
Judging the parity of a function according to its definition;
A, firstly, determining the definition domain of a function and judging whether it is symmetrical about the origin;
B. determine the relationship between f(-x) and f(x);
C, draw the corresponding conclusion: if f(-x)=f(x) or f(-x)-f(x)=0, then f(x) is an even function; If f(-x)=-f(x) or f(-x)+f(x)=0, then f(x) is odd function.
Note: Symmetry of the definition domain of a function about the origin is a necessary condition for the function to have parity. First, whether the domain of the function is symmetric about the origin, and if not, whether the function is odd or even. If it is symmetric, (1) will be judged according to the definition. (2) judging by f (-x) f (x) = 0 or f (x)/f (-x) = 1; (3) Using the image judgment of theorems or functions.
9. Analytic expression of function
(1). The analytic formula of the function is a representation of the function. When the functional relationship between two variables is needed, the corresponding law between them and the definition domain of the function are needed.
(2) The main methods for finding analytic expressions of functions are:
1) matching method
2) undetermined coefficient method
3) Alternative methods
4) Parameter elimination method
10. Maximum (minimum) value of the function (see textbook p36 for definition).
A. Use the property of quadratic function (collocation method) to find the maximum (minimum) value of the function.
B. Use the image to find the maximum (minimum) value of the function.
C, judging the maximum (minimum) value of the function by using the monotonicity of the function;
If the function y=f(x) monotonically increases in the interval [a, b] and monotonically decreases in the interval [b, c], then the function y=f(x) has the maximum value f (b) at x=b;
If the function y=f(x) monotonically decreases in the interval [a, b] and monotonically increases in the interval [b, c], then the function y=f(x) has a minimum value f (b) at x=b; .
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