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Induction of knowledge points of function nature in senior one.
I believe that the knowledge discovered by human beings will only flow to those who need it. To some extent, man is only the carrier of knowledge, and knowledge is a special energy that can be produced and consumed. Let's share some knowledge points about the nature of senior one function, hoping to help you.

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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