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.

X is an independent variable and y is a function of X.

[Edit this paragraph] Three expressions of quadratic function

① general formula: y = ax 2+bx+c (a, b and c are constants, a≠0).

② Vertex [vertex P(h, k) of parabola]: y = a (x-h) 2+k.

③ Intersection point [only applicable to parabolas with intersection points A(x 1 0) and B(x2, 0) with the X axis ]: y = a (x-x 1) (x-x2).

The above three forms can be converted as follows:

Relationship between (1) General Formula and Vertex Type

For the quadratic function y = ax 2+bx+c, its vertex coordinates are (-b/2a, (4ac-b 2)/4a), that is.

h=-b/2a=(x 1+x2)/2

k=(4ac-b^2)/4a

② Relationship between general formula and intersection point

X 1, x2 = [-b √ (b 2-4ac)]/2a (that is, the formula for finding the root of a quadratic equation with one variable).

[Edit this paragraph] image of quadratic function

The image of quadratic function y = x 2 in plane rectangular coordinate system,

It can be seen that the image of quadratic function is an endless parabola.

[Edit this paragraph] The properties of parabola

1. Parabola is an axisymmetric figure. The symmetry axis 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 coordinate is 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 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

When δ = b 2-4ac > 0, the parabola has two intersections with the X axis.

When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.

_______

When δ = b 2-4ac < 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).

When a>0, the function obtains the minimum value f (-b/2a) = 4ac-b2/4a at x= -b/2a; In {x | x-b/2a} is an increasing function; The opening of parabola is upward; The range of the function is {y | y ≥ 4ac-b 2/4a}, and vice versa.

When b=0, the axis of symmetry of parabola is the Y axis. At this point, the function is even, and the analytical expression is transformed into y = ax 2+c (a ≠ 0).

7. domain: r

Range: ① [(4ac-b 2)/4a, positive infinity]; ②[t, positive infinity]

Parity: even function

Periodicity: None

Analytical formula:

①y = ax2+bx+c[ general formula]

⑴a≠0

(2) when a > 0, the parabolic opening is upward; A < 0, parabolic opening downward;

⑶ Extreme point: (-b/2a, (4ac-b2)/4a);

⑷δ=b^2-4ac，

δ> 0, where the image intersects the X axis at two points:

([-b+√ δ]/2a, 0) and ([-b+√δ]/2a, 0);

Δ = 0, the image intersects the x axis at one point:

(-b/2a，0)；

δ < 0, the image has no intersection with the X axis;

②y = a(x-h)2+t[ collocation]

The corresponding extreme point is (h, t), where h=-b/2a and t = (4ac-b2)/4a);

[Edit this paragraph] Quadratic function and unary quadratic equation

In particular, the quadratic function (hereinafter called 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.

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.

Analytical formula

y=ax^2

y=a(x-h)^2

y=a(x-h)^2+k

y=ax^2+bx+c

Vertex coordinates

(0,0)

(h，0)

(h，k)

(-b/2a,sqrt[4ac-b^2]/4a)

axis of symmetry

x=0

x=h

x=h

x=-b/2a

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 h < 0, k >; 0, moving the parabola to the left by |h| units in parallel, and then moving it up by k units to obtain 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？ In addition, the distance between any pair of symmetrical points on the parabola can be | 2× (-b/2a)-a | (A is the abscissa of a point).

When △ = 0, the image has only one intersection with the X axis;

When delta < 0. The image does not intersect with the x axis. When a >; 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)。

7. The knowledge of quadratic function can be easily integrated with other knowledge, resulting in more complex synthesis problems. Therefore, the comprehensive question based on quadratic function knowledge is a hot topic in the senior high school entrance examination, which often appears in the form of big questions.