Vertex: y=a(x-h)? 0? 5+k or y=a(x+m)? 0? 5+k (the two formulas are essentially the same, but the junior high school textbooks are both the first formula).
Intersection point (with x axis): y=a(x-x 1)(x-x2)
Important concepts: (a, b, c are constants, a≠0, a determines the opening direction of the function, a >；; 0, the opening direction is upward, a
The right side of a quadratic function expression is usually quadratic.
X is an independent variable and y is a quadratic function of X.
X 1, x2 = [-b (b 2-4ac) under the root sign ]/2a (that is, the formula for finding the root of a quadratic equation with one variable).
[Edit this paragraph] image of quadratic function
Make the square of quadratic function y=x in the plane rectangular coordinate system; Images of,
It can be seen that the image of quadratic function is an endless parabola. Different quadratic function images
[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. A parabola has a vertex p with coordinates P (-b/2a, (4ac-b? 0? 5)/4a)
-b/2a=0, p is on the y axis; When δδ= b? 0? When 5-4ac=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; Because if the axis of symmetry is on the left, the axis of symmetry is less than 0, which is -b/2a.
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. Because the axis of symmetry is on the right, the axis of symmetry is greater than 0, that is,-b/2a >; 0, so b/2a should be less than 0, so a and b should have different signs.
It can be simply recorded as the same as left and right, that is, when the symbols 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.
In fact, b has its own geometric meaning: the value of the slope k of the analytic function (linear function) of the parabola tangent at the intersection of parabola and Y axis. It can be obtained by taking the derivative of quadratic function.
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？ 0? When 5-4ac > 0, there are two intersections between parabola and X axis.
δ= b？ 0? When 5-4ac=0, the parabola has 1 intersection points with the X axis.
_______
δ= b？ 0? When 5-4ac < 0, the parabola has no intersection with the X axis. The value of x is an imaginary number (x =-b √ b? 0? The reciprocal of the value of 5-4ac is multiplied by the imaginary number i, and the whole equation is divided by 2a).
When a>0, the function obtains the minimum value f(-b/2a)=4ac-b at x= -b/2a? 0? 5/4a； In {x | x-b/2a} is an increasing function; The opening of parabola is upward; The range of the function is {y | y ≥ 4ac-b2; /4a} On the contrary, it remains unchanged.
When b=0, the axis of symmetry of parabola is the Y axis. At this point, the function is an even function, and the analytical expression is deformed into y=ax? 0? 5+c(a≠0)
7. domain: r
Scope: (Corresponding to the analytical formula, and only discussing the case that A is greater than 0, please ask the reader to infer the case that A is less than 0) ①[(4ac-b? 0? 5)/4a, positive infinity); ②[t, positive infinity]
Parity: even function
Periodicity: None
Analytical formula:
①y=ax？ 0? 5+bx+c[ general formula]
⑴a≠0
(2) when a > 0, the parabolic opening is upward; A < 0, parabolic opening downward;
(3) Extreme point: (-b/2a, (4ac-b? 0? 5)/4a)；
⑸δ= b？ 0? 5-4ac，
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)？ 0? 5+t[ collocation method]
At this time, the corresponding extreme point is (h, t), where h=-b/2a and t=(4ac-b? 0? 5)/4a)；
③y=a(x-x 1)(x-x2)[ intersection]
A≠0, where x 1 and x2 are the two intersections of the function and the x axis, and the analytical formula can be obtained by substituting x and y (usually using a quadratic equation).
[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； The images of +k, y = ax 2+bx+c (a≠0 in all kinds) have the same shape, but different positions. Their vertex coordinates and symmetry axes are as follows:
Analytical formula
y=ax^2；
y=ax^2； +K
y=a(x-h)^2；
y=a(x-h)^2+k
y=ax^2+bx+c
Vertex coordinates
(0,0)
(0，K)
(h，0)
(h，k)
(-b/2a,sqrt[4ac-b^2； ]/4a)
axis of symmetry
x=0
x=0
x=h
x=h
x=-b/2a
When h>0, y = a (x-h) 2; The image can be represented by parabola y = ax 2; Move the h unit in parallel to the right,
When h < 0, it is obtained by moving |h| units in parallel to the left.
When h>0, k>0 and parabola y = ax 2; Move H units in parallel to the right, and then move K units upward, and you can get an image of y = a (x-h) 2+k;
When h>0, k<0 and parabola y = ax 2; An image with y = a (x-h) 2-k can be obtained by moving h units in parallel to the right and then moving down | k units;
When h < 0, k >; 0, move the parabola to the left by |h| units in parallel, and then move it up by k units to get y=a(x+h)? 0? 5+k image;
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 get y=a(x-h)? 0? 5+k image;
Therefore, the image of parabola Y = AX 2+BX+C (A ≠ 0) is studied, and the general formula is changed to Y = A (X-H) 2 through the formula; In the form of +k, its vertex coordinates, symmetry axis and approximate position of parabola can be clearly determined, 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? 6? 9,0) and B(x? 6? 0,0), where x 1, x2 is the unary quadratic equation ax 2+bx+c = 0.
(a≠0)。 The distance between these two points AB=|x? 6? 0-x？ 6? 9| 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 known vertex coordinate or symmetry axis or the maximum (minimum) value of the 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? 6? 9)(x-x？ 6? 0)(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.