I. Definition and definition of expressions
Generally speaking, there is the following relationship between independent variable x and dependent variable y:
y=ax？ +bx+c(a, b, c are constants, a≠0)
Y is called the quadratic function of X.
The right side of a quadratic function expression is usually a quadratic trinomial.
Two. Three Expressions of Quadratic Function
General formula: y=ax? +bx+c(a, b, c are constants, a≠0)
Vertex: y=a(x-h)? +k[ vertex P(h, k) of parabola]
Intersection point: y = a(X-X 1)(X-x2)[ only applicable to parabolas with intersection points a (x 1, 0) and b (x2, 0) with the x axis]
Note: Among these three forms of mutual transformation, there are the following relations:
h=-b/2a k=(4ac-b？ )/4a x 1，x2=(-b √b？ -4ac)/2a
Three. Image of quadratic function
Do quadratic function y=x in plane rectangular coordinate system? Images,
It can be seen that the image of quadratic function is a parabola.
Four. Properties of parabola
1. Parabola is an axisymmetric figure. The axis of symmetry is a straight line
x = -b/2a .
The only intersection of the symmetry axis and the parabola is the vertex p of the parabola.
Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).
2. The parabola has a vertex p, and the coordinates are
P [ -b/2a，(4ac-b？ )/4a ].
-b/2a=0, p is on the y axis; When δδ= b? When -4ac=0, p is on the x axis.
3. Quadratic coefficient A determines the opening direction and size of parabola.
When a > 0, the parabola opens upward; When a < 0, the parabola opens downward.
The larger the |a|, the smaller the opening of the parabola.
4. Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.
When the signs of A and B are the same (that is, AB > 0), the symmetry axis is left on the Y axis;
When the signs of A and B are different (that is, AB < 0), the symmetry axis is on the right side of the Y axis.
5. The constant term c determines the intersection of parabola and Y axis.
The parabola intersects the Y axis at (0, c)
6. Number of intersections between parabola and X axis
δ= b？ When -4ac > 0, the parabola has two intersections with the x-axis.
δ= b？ When -4ac=0, the parabola has 1 intersections with the X axis.
δ= b？ When -4ac < 0, the parabola has no intersection with the x axis.
Verb (abbreviation of verb) quadratic function and unary quadratic equation
Especially quadratic function (hereinafter referred to as function) y=ax? +bx+c，
When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).
Is that an axe? +bx+c=0
At this point, whether the function image intersects with the X axis means whether the equation has real roots.
The abscissa of the intersection of the function and the x axis is the root of the equation.
linear function
I. Definitions and definitions:
Independent variable x and dependent variable y have the following relationship:
Y=kx+b(k, b is a constant, k≠0)
It is said that y is a linear function of x.
In particular, when b=0, y is a proportional function of x.
Two. Properties of linear functions:
The change value of y is directly proportional to the corresponding change value of x, and the ratio is K.
That is △ y/△ x = K.
Three. Images and properties of linear functions;
1. exercises and graphics: through the following three steps (1) list; (2) tracking points; (3) Connecting lines can make images of linear functions-straight lines. So the image of a function only needs to know two points and connect them into a straight line.
2. Property: any point P(x, y) on the linear function satisfies the equation: y = kx+b.
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 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 one or three quadrants; When k < 0, the straight line only passes through two or four quadrants.
Four. Determine the expression of 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, two equations can be listed:
Y 1 = KX 1+B 1，Y2 = KX2+B2。
(3) Solve this binary linear equation and get the values of K and B. ..
(4) Finally, the expression of the linear function is obtained.
The application of verb (verb's abbreviation) linear function in life
1. When the time t is constant, the distance s is a linear function of the velocity v .. s=vt.
2. When the pumping speed f of the pool is constant, the water quantity g in the pool is a linear function of the pumping time t, and the original water quantity s in the pool is set. G = S- feet.
inverse proportion function
A function in the form of y = k/x (where k is a constant and k≠0) is called an inverse proportional function.
The range of the independent variable x is all real numbers that are not equal to 0.
The image of the inverse proportional function is a hyperbola.
As shown in the figure, the function images when k is positive and negative (2 and -2) are given above.
trigonometric function
Trigonometric function is a kind of transcendental function in elementary function in mathematics. Their essence is the mapping between the set of arbitrary angles and a set of ratio variables. The usual trigonometric function is defined in the plane rectangular coordinate system, and its domain is the whole real number domain. The other is defined in a right triangle, but it is incomplete. Modern mathematics describes them as the limit of infinite sequence and the solution of differential equation, and extends their definitions to complex system.
Because of the periodicity of trigonometric function, it does not have the inverse function in the sense of single-valued function.
Trigonometric functions have important applications in complex numbers. Trigonometric function is also a common tool in physics.
It has six basic functions:
Function name sine cosine tangent cotangent secant cotangent
Symbol sin cos tan cot sec csc
Sine function sin(A)=a/h
Cosine function cos(A)=b/h
Tangent function tan(A)=a/b
Cotangent function cot(A)=b/a
In a certain change process, the two variables X and Y, for each value of X within a certain range, Y has a certain value corresponding to it, and Y is a function of X. This relationship is generally expressed by y=f(x).