linear function
Definition and expression of 1. linear function;
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)
2. 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.
3. Images and properties of linear functions;
(1) Exercise and Graphics: Through the following three steps.
A list;
B. Tracking points;
C line can be an 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) nature:
Any point P(x, y) of a on a linear function satisfies the equation: y = kx+b.
The coordinate of the intersection of the linear function of b and the Y axis is always (0, b), and the image of the proportional function of (-b/k, 0) always crosses the origin.
(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.
4. 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, 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.
5. The application of 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 ... Set the original water quantity in the pool. G = S- feet.
6. Commonly used formula:
(1) Find the k value of the function image: (y 1-y2)/(x 1-x2).
(2) Find the midpoint of the line segment parallel to the X axis: |x 1-x2|/2.
(3) Find the midpoint of the line segment parallel to the Y axis: |y 1-y2|/2.
(4) Find the length of any line segment: square sum of √ (x 1-x2)' 2+(y 1-y2)' 2 (note: under the root sign, (x1-x2) and (y1-y2).
quadratic function
1. Define and define expressions.
Generally speaking, there is the following relationship between independent variable x and dependent variable y:
y=ax'2+bx+c
(a, b, c are constants, a≠0, a determines the opening direction of the function, a >；; 0, the opening direction is upward, a
Y is called the quadratic function of X.
The right side of a quadratic function expression is usually a quadratic trinomial.
2. Three expressions of quadratic function
General formula: y=ax'2+bx+c(a, b and c are constants, and a≠0).
Vertex: y=a(x-h)'2+k[ vertex P(h, k) of parabola]
Intersection point: y=a(x-x? )(x-x？ ) [only when it is related to the x axis A(x? , 0) and B(x? 0) parabola]
Note: Among these three forms of mutual transformation, there are the following relations:
h=-b/2ak=(4ac-b'2)/4ax？ ，x？ =(-b √b'2-4ac)/2a
3. Quadratic function image
Make an image of quadratic function y = x'2 in a plane rectangular coordinate system,
It can be seen that the image of quadratic function is a parabola.
4. The 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) A parabola has a vertex p with coordinates.
P(-b/2a，(4ac-b'2)/4a)
-b/2a=0, p is on the y axis; When δ = b' 2-4ac = 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 upwards; 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 A and B have the same number (ab>0), the symmetry axis is on the left side of Y axis;
When a and b have different numbers (i.e. AB
(5) The constant term c determines the intersection of the parabola and the Y axis.
The parabola intersects the Y axis at (0, c)
(6) Number of intersections between parabola and X axis
δ= b ' 2-4ac & gt； 0, parabola and x axis have two intersections.
When δ = b' 2-4ac = 0, the parabola has 1 intersection points with the X axis.
δ= b ' 2-4ac & lt； 0, the parabola has no intersection with the x axis. The value of x is an imaginary number (the reciprocal of the value of x =-b √ b' 2-4ac, multiplied by the imaginary number I, and the whole formula is divided by 2a).
5. Quadratic function and unary quadratic equation
Particularly, the quadratic function (hereinafter referred to as function) y = ax'2+bx+c,
When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).
That is, ax'2+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.
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