Summary of knowledge points of quadratic function in junior high school mathematics I. Definition and definition expression
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
The right side of a quadratic function expression is usually a quadratic trinomial.
Two. 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/2a k=(4ac-b^2)/4a x？ ，x？ =(-b √b^2-4ac)/2a
Three. Quadratic function image
Making the image of quadratic function y = x 2 in plane rectangular coordinate system, we can see that the image of quadratic function is parabola.
Four. 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 its coordinates are: P(-b/2a, (4ac-b 2)/4a) When -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 parabola and Y axis.
The parabola intersects the Y axis at (0, c)
6. Number of intersections between parabola and X axis
δ=b^2-4ac>； 0, parabola and x axis have two intersections.
When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.
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).
Verb (abbreviation of verb) 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.
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？ |
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.
Extended reading: junior high school mathematics learning methods 1, preview reading before class. When previewing the text, you should prepare a piece of paper and a pen, and write down the key words, questions and problems that need to be considered in the textbook. You can simply repeat and reason about definitions, axioms, formulas and rules on paper. Key knowledge can be approved, marked, circled and marked in textbooks. Doing so not only helps us to understand the text, but also helps us to concentrate on listening in class.
2. Read books in class. When previewing, we only have a general understanding of the contents of the textbooks to be learned, and not all of them have been thoroughly understood and digested. Therefore, it is necessary to read the text further in combination with the marks and comments made in the preview and the teacher's teaching, so as to grasp the key points and solve the difficult problems in the preview.
3. Review reading after class. After-class review is an extension of classroom learning, which can not only solve the unresolved problems in preview and classroom, but also systematize knowledge, deepen and consolidate the understanding and memory of classroom learning content. After a class, you must read the textbook first, and then do your homework. After learning a unit, you should read the textbook comprehensively, connect the content of this unit before and after, summarize it comprehensively, write a summary of knowledge, and check for missing parts.