The range of the independent variable x is all real numbers that are not equal to 0.
Inverse proportional function image properties:
The image of the inverse proportional function is a hyperbola.
Since the inverse proportional function belongs to odd function, let f(-x)=-f(x), and the image is symmetrical about the origin.
In addition, from the analytical formula of inverse proportional function, it can be concluded that any point on the image of inverse proportional function is perpendicular to two coordinate axes, and the rectangular area surrounded by this point, two vertical feet and the origin is a constant, which is ∣k∣.
When K>0, the inverse proportional function image passes through one or three quadrants, which is a decreasing function (that is, Y decreases with the increase of X).
When k < 0, the inverse proportional function image passes through two or four quadrants, which is increasing function (that is, y increases with the increase of x).
Because the independent variable and dependent variable of the inverse proportional function cannot be zero, the image can only approach the coordinate axis infinitely, and cannot intersect the coordinate axis.
1. Any point on the inverse proportional function image is a vertical line segment of two coordinate axes, and the area of the rectangle surrounded by these two vertical line segments and coordinate axes is |k|.
2. For hyperbola y=k/x, if you add or subtract any real number on the denominator (that is, y = k/x (x m) m is constant), it is equivalent to translating the hyperbola image to the left or right by one unit. (When adding a number, move to the left, and when subtracting a number, move to the right)
Chapter 2: Quadratic function.
Knowledge point 1. rectangular coordinates/ cartesian coordinates
1, plane rectangular coordinate system
Draw two mutually perpendicular number axes with a common origin on the plane to form a plane rectangular coordinate system.
Among them, the horizontal axis is called X axis or horizontal axis, and the right direction is the positive direction; The vertical axis is called Y axis or vertical axis, and the orientation is positive; The intersection o of the two axes (that is, the common * * *) is called the origin of the rectangular coordinate system; The plane on which the rectangular coordinate system is established is called the coordinate plane.
In order to describe the position of a point in the coordinate plane conveniently, the coordinate plane is divided into four parts, namely the first quadrant, the second quadrant, the third quadrant and the fourth quadrant.
Note: The points on the X and Y axes do not belong to any quadrant.
2, the concept of point coordinates
The coordinates of points are represented by (a, b), and the order is abscissa before, ordinate after, and there is a ","in the middle. The positions of horizontal and vertical coordinates cannot be reversed. The coordinates of points on the plane are ordered real number pairs. At that time, (a, b) and (b, a) were the coordinates of two different points.
Knowledge point 2. Coordinate characteristics of different position points
1, the coordinate characteristics of the midpoint of each quadrant.
Point P(x, y) is in the first quadrant.
Point P(x, y) is in the second quadrant.
Point P(x, y) is in the third quadrant.
Point P(x, y) is in the fourth quadrant.
2. Features of points on the coordinate axis
The point P(x, y) is on the X axis, and X is an arbitrary real number.
The point P(x, y) is on the y axis, and y is an arbitrary real number.
Point P(x, y) is on both X and Y axes, and both X and Y are zero, that is, the coordinate of point p is (0,0).
3. Characteristics of the coordinates of points on the bisector of two coordinate axes.
Point P(x, y) is equal to x on the bisector of the first and third quadrants.
Points P(x, y) are opposite to each other on the bisector of the second and fourth quadrants.
4. Coordinate characteristics of points on a straight line parallel to the coordinate axis
The ordinate of each point on the straight line parallel to the X axis is the same.
The abscissa of each point on the straight line parallel to the Y axis is the same.
5. Coordinate characteristics of points symmetrical about X-axis, Y-axis or apogee.
The abscissa of point P and point P' is equal to the axis of X, and the ordinate is opposite.
The vertical coordinates of point P and point P' are symmetrical about the Y axis, and the horizontal coordinates are opposite to each other.
Point p and point p' are symmetrical about the origin, and the abscissa and ordinate are opposite.
6. Distance from point to coordinate axis and origin
Distance from point P(x, y) to coordinate axis and origin:
(1) The distance from the point P(x, y) to the X axis is equal to
(2) The distance from the point P(x, y) to the Y axis is equal to
(3) The distance from point P(x, y) to the origin is equal to
Chapter 3: The image and properties of quadratic function.
Concept of quadratic function: Generally speaking, a function with the shape of AX 2+BX+C = 0 is called a quadratic function.
What needs to be emphasized here is that, similar to the unary quadratic equation, the coefficient of the quadratic term is a≠0, while b and c can be zero. The domains of quadratic functions are all real numbers.
Quadratic function image and its property formula
Quadratic parabola, image symmetry is the key;
Define the opening, vertex and intersection of the image boundary;
The opening and size are broken by A, C intersects the Y axis, and the symbol of B is special, and the symbol is associated with A; First find the vertex position, take the y axis as the baseline, and the difference between the left and right is 0, remember that there is no doubt in your heart; Vertex coordinates are the most important and appear in the general formula. The horizontal scale is the axis of symmetry, and the vertical scale function is the most important. If the position of the symmetry axis is found, the sign will be reversed and different expressions can be interchanged.
Chapter four: the image of function and quadratic equation of one variable.
1. quadratic function y = ax 2, Y = A (X-H) 2, Y = A (X-H) 2+K, y = ax 2+bx+c (among all kinds, a≠0) has the same image shape, but different positions.
When h>0, the parabola y = ax 2 is moved to the right by H units in parallel, and the image of y = a (x-h) 2 can be obtained.
When h < 0, it is obtained by moving |h| units in parallel to the left.
When h>0, k>0, the parabola y = ax 2 is moved to the right by H units in parallel, and then moved up by K units, the image of y = a (x-h) 2+k can be obtained;
When h>0, k<0, the parabola y = ax 2 is moved to the right by h units in parallel, and then moved down by | k units, and the image of y = a (x-h) 2+k is obtained;
When h0, move the parabola to the left by |h| units in parallel, and then move it up by K units to get an image with y = a (x-h) 2+k;
When h < 0, k<0, move the parabola to the left by |h| units in parallel, and then move it down by |k| units to obtain an image with y = a (x-h) 2+k;
Therefore, it is very clear to study the image of parabola y = ax 2+bx+c (a ≠ 0) and change the general formula into the form of Y = A (X-H) 2+K through the formula, so as to determine its vertex coordinates, symmetry axis and approximate position of parabola, which provides convenience for drawing images.
2. the image of parabola y = ax 2+bx+c (a ≠ 0): when a >: 0, the opening is upward, when a.
3. parabola y = ax 2+bx+c (a ≠ 0), if a >；; 0, when x≤-b/2a, y decreases with the increase of x; When x≥-b/2a, y increases with the increase of x, if a
4. The intersection of the image with parabola y = ax 2+bx+c and the coordinate axis:
(1) The image must intersect with the Y axis, and the coordinate of the intersection point is (0, c);
(2) when △ = b 2-4ac >; 0, the image intersects the x axis at two points A(x? , 0) and B(x? 0), where x 1, x2 is the unary quadratic equation ax 2+bx+c = 0.
(a≠0)。 The distance between these two points AB=|x? -x？ |
When △=0, the image has only one intersection with the X axis;
When △0, the image falls above the X axis, and when X is an arbitrary real number, there is y>0; When a<0, the image falls below the X axis, and when X is an arbitrary real number, there is Y.
5. the maximum value of parabola y = ax 2+bx+c: if a>0 (a <; 0), then when x=-b/2a, the minimum (large) value of y = (4ac-b 2)/4a.
The abscissa of the vertex is the value of the independent variable when the maximum value is obtained, and the ordinate of the vertex is the value of the maximum value.
6. Find the analytic expression of quadratic function by undetermined coefficient method.
(1) When the given condition is that the known image passes through three known points or three pairs of corresponding values of known x and y, the analytical formula can be set to the general form:
y=ax^2+bx+c(a≠0).
(2) When the given condition is the vertex coordinate or symmetry axis of the known image, the analytical formula can be set as the vertex: y = a (x-h) 2+k (a ≠ 0).
(3) When the given condition is that the coordinates of two intersections between the image and the X axis are known, the analytical formula can be set as two formulas: y=a(x-x? )(x-x？ )(a≠0)。
Chapter 5: The application of quadratic function.
Roads, bridges, tunnels, urban construction and many other aspects have parabolas; In production and life, there are many problems such as "profit", "least materials", "least expenses", "shortest route" and "area", all of which may use the relationship of quadratic function and the maximum value of quadratic function.
Then the general steps to solve this kind of problem are:
Step 1: Set independent variables;
Step 2: establish a resolution function;
Step 3: Determine the range of independent variables;
Step 4: Find the maximum value (within the range of independent variables) according to the vertex coordinate formula or matching method.