Induction of knowledge points of mathematical quadratic function
Calculation method
1. Average sample: (1); (2) If,,,,,, then (A- constant,,,, is close to a more integer constant A); (3) Weighted average: (4) Average is a characteristic number that describes the trend (concentration position) in data concentration. Sample average is usually used to estimate the overall average. The larger the sample size, the more accurate the estimation.
2. Sample variance: (1); (2) If,,,, then (a- an "integer" constant closer to the average value of,,,); If … is less than "whole", then; ⑶ Sample variance is a characteristic number that describes the degree of data dispersion (fluctuation). When the sample size is large, the sample variance is very close to population variance, which is usually used to estimate population variance.
3. Sample standard deviation:
Third, the application examples (omitted)
Mathematics knowledge points of grade three: chapter four linear form
★ Emphasis★ Concepts, judgments and properties of intersecting lines and parallel lines, triangles and quadrangles.
☆ Summary ☆
I. Straight lines, intersecting lines and parallel lines
1. Differences and connections between line segments, rays and straight lines
This paper analyzes the graph, representation, boundary, number of endpoints and basic properties.
2. The midpoint of the line segment and its representation
3. Basic properties of straight lines and line segments (using "basic properties of line segments" to demonstrate that "the sum of two sides of a triangle is greater than the third side")
4. The distance between two points (three distances: point-point; Dotted line; Line-line)
5. Angle (flat angle, rounded corner, right angle, acute angle, obtuse angle)
6. Complementary angle, complementary angle and their expressions
7. The bisector of an angle and its representation
8. Vertical line and its basic properties (use it to prove that "the hypotenuse of a right triangle is greater than the right")
9. Vertex angle and its properties
10. Parallel lines and their judgments and properties (reciprocal) (differences and connections between them)
1 1. Common theorems: ① parallel to two straight lines and parallel to one straight line (transitivity); ② Two straight lines parallel to and perpendicular to a straight line.
12. Definition, proposition and composition of proposition
13. Axioms and theorems
14. Inverse proposition
Second, the triangle
Classification: (1) Classification by edge;
(2) according to the angle.
1. Definition (including internal angle and external angle)
2. The relationship between angles of triangle: (1) the sum and inference of angles and angles: (1) inner angles; ② sum of external angles; (3) the sum of the internal angles of the N-polygon; (4) the sum of the external angles of the N-polygon. ⑵ Edge and edge: The sum of two sides of a triangle is greater than the third side, and the difference between the two sides is less than the third side. ⑶ Angle and edge: In the same triangle,
3. The main part of the triangle
Discussion: ① Define the intersection of ② _ _ lines-the property of triangle× center ③.
① High line ② Middle line ③ Angle bisector ④ Middle vertical line ⑤ Middle line.
⑵ General triangle ⑵ Special triangle: right triangle, isosceles triangle and equilateral triangle.
4. Determination and properties of special triangles (right triangle, isosceles triangle, equilateral triangle and isosceles right triangle)
5. congruent triangles
(1) Determine the consistency of general triangles (SAS, ASA, AAS, SSS).
⑵ Determination of congruence of special triangle: ① General method ② Special method.
6. Area of triangle
⑴ General calculation formula ⑴ Properties: The areas of triangles with equal bases and equal heights are equal.
7. Important auxiliary lines
(1) The midpoint and the midpoint form the midline; (2) Double the center line; (3) Add auxiliary parallel lines
8. Proof method
(1) direct proof method: synthesis method and analysis method.
(2) Indirect proof-reduction to absurdity: ① Counterhypothesis ② Reduction to absurdity ③ Conclusion.
(3) Prove that line segments are equal and angles are equal, often by proving triangle congruence.
(4) Prove the folding relationship of line segments: folding method and folding method.
5. Prove the sum-difference relationship of line segments: continuation method and truncation method.
[6] Prove the area relationship: indicate the area.
Third, quadrilateral.
Classification table:
1. General Properties (Angle)
⑴ Sum of internal angles: 360.
(2) Parallelogram connecting the midpoint of each side in turn.
Inference 1: Connect the midpoints of the sides of the quadrilateral in turn with equal diagonal lines to get a diamond.
Inference 2: Connect the midpoints of the sides of the quadrilateral in turn with diagonal lines perpendicular to each other to get a rectangle.
⑶ Sum of external angles: 360.
2. Special quadrilateral
(1) General methods to study them:
(2) parallelogram, rectangle, diamond and square; Definition, properties and judgment of trapezoid and isosceles trapezoid
⑶ Determination steps: quadrilateral → parallelogram → rectangle → square.
┗→ Diamonds-=
(4) diagonal tie rod:
3. Symmetric graphics
(1) axis symmetry (definition and properties); (2) Central symmetry (definition and nature)
4. Related Theorems: ① Parallel bisection theorem and its inference 1, 2.
② The midline theorem of triangle and trapezoid.
③ The distance between parallel lines is equal everywhere. (For example, find triangles with equal areas in the figure below)
5. Important auxiliary lines: ① Diagonal lines of quadrangles are often connected; ② Trapezoids are often transformed into triangles by translating a waist, translating a diagonal, making a height, connecting the midpoint between the vertex and the waist and extending the intersection with the bottom.
6. Drawing: Divide the line segments randomly.
Summary of knowledge points of quadratic function
I. Definition and definition of 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
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.
Summary of knowledge points of quadratic function
Quadratic function concept
Generally speaking, a shape like y=ax? The function of +bx+c (where a, b and c are constants and a≠0, b and c can be 0) is called quadratic function, where a is called quadratic coefficient, b is linear coefficient and c is a constant term. X is the independent variable and y is the dependent variable. The maximum number of independent variables to the right of the equal sign is 2. Quadratic function image is an axisymmetric figure.
Note: "variable" is different from "independent variable", so it cannot be said that "quadratic function refers to a polynomial function with the highest degree of a variable being quadratic". "Unknown" is just a number (the specific value is unknown, but only one value is taken), and "variable" can take any value within the real number range. The concept of "unknown" is applied in the equation (both functional equation and differential equation are unknown functions, but both unknown and unknown functions generally represent a number or function-special circumstances may occur), but the letters in the function represent variables and their meanings have always been different. From the definition of function, we can also see the difference between them, just as function is not equal to function.
Complete works of quadratic function formulas
quadratic function
I. Definition and definition of expressions
Generally speaking, there is the following relationship between independent variable x and dependent variable y:
y=ax？ +bx+c(a, b, c are constants, a≠0)
Y is called the quadratic function of X.
The right side of a quadratic function expression is usually a quadratic trinomial.
Two. Three Expressions of Quadratic Function
General formula: y=ax? ; +bx+c(a, b, c are constants, a≠0)
Vertex: y=a(x-h)? ; +k[ vertex P(h, k) of parabola]
Intersection point: y = a(X-X 1)(X-x2)[ only applicable to parabolas with intersection points a (x 1, 0) and b (x2, 0) with the x axis]
Note: Among these three forms of mutual transformation, there are the following relations:
h=-b/2a k=(4ac-b？ ; )/4a x 1，x2=(-b √b？ ; -4ac)/2a
Three. Image of quadratic function
Let the quadratic function y=x be the image in the plane rectangular coordinate system,
It can be seen that the image of quadratic function is a parabola.
Four. 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. The parabola has a vertex p, and the coordinates are
P [ -b/2a，(4ac-b？ ; )/4a ].
-b/2a=0, p is on the y axis; When δδ= b? When -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？ -4ac & gt； 0, parabola and x axis have two intersections.
δ= b？ When -4ac=0, the parabola has 1 intersections with the X axis.
δ= b？ -4ac & lt； 0, the parabola has no intersection with the x axis.
Verb (abbreviation of verb) quadratic function and unary quadratic equation
Especially quadratic function (hereinafter referred to as function) y=ax? ; +bx+c，
When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).
Is that an axe? ; +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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