(20 12-03- 18 09: 10:26)
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education
Knowledge point 1. concept
Figures with the same shape are called similar figures. (i.e. graphs with equal corresponding angles and equal corresponding edge ratios)
Interpretation: (1) Two graphs are similar, and one graph can be seen as being enlarged or reduced by the other graph.
(2) Conformity can be regarded as a special similarity, that is, not only the same shape, but also the same size.
(3) Judging whether two figures are similar depends on whether they are the same shape, which has nothing to do with other factors.
Knowledge point 2. Proportional line segment
For four line segments A, B, C and D, if the ratio of the lengths of two of them is equal to the ratio of the lengths of the other two, that is (or a:b=c:d), then these four line segments are called proportional line segments.
Knowledge point 3. Properties of similar polygons
Properties of similar polygons: the corresponding angles of similar polygons are equal, and the proportions of corresponding edges are equal.
Interpretation: (1) Understand the definition of similar polygons correctly and make clear the corresponding relationship.
(2) It is clear that the correspondence of similar polygons comes from writing, and the similarity ratio is sequential.
Knowledge point 4. Similar triangles's concept
A triangle with equal corresponding angles and equal ratio of corresponding sides is called similar triangles.
Interpretation: (1) similar triangles is one of the similar polygons;
(2) similar triangles should be understood by combining the properties of similar polygons;
(3) similar triangles should have the same shape, but different sizes;
(4) Similarity is indicated by "√" and pronounced as "similar to";
(5) The ratio of corresponding sides in similar triangles is called similarity ratio.
Knowledge point 5. Similar triangles's judgment method
(1) Definition: Two triangles with equal corresponding angles and proportional corresponding sides are similar;
(2) The triangle formed by cutting the other two sides (or extension lines of other two sides) with a straight line parallel to one side of the triangle is similar to the original triangle.
(3) If two angles of a triangle are equal to two angles of another triangle, then the two triangles are similar.
(4) If two sides of a triangle are proportional to two sides of another triangle, and the included angles are equal, then the two triangles are similar.
(5) If three sides of a triangle are proportional to three sides of another triangle, then the two triangles are similar.
(6) Two right triangles divided by the height on the hypotenuse are similar to the original triangle.
Knowledge point 6. The nature of similar triangles
(1) The corresponding angles are equal, and the ratio of the corresponding sides is equal;
(2) The ratio corresponding to the height, the ratio corresponding to the center line and the ratio corresponding to the angular bisector are all equal to the similarity ratio;
(3) The ratio of similar triangles perimeter is equal to the similarity ratio; The area ratio is equal to the square of the similarity ratio.
(4) Projective theorem
Definition of central symmetry
Rotate the figure around a point 180. If it can coincide with another figure, it is said that the two figures are symmetrical or symmetrical about this point, which is called the symmetrical center, and the corresponding points of the two figures are called the symmetrical points about the center. The relationship between central symmetry and central symmetry figure
Central symmetry and central symmetry figure are two different but closely related concepts. The difference is that central symmetry refers to the mutual positional relationship between two congruent figures. These two figures are symmetrical about a point, which is the center of symmetry, and the symmetry of the two figures about a point is also called central symmetry. In two graphs with central symmetry, the symmetry points of all points on one graph are on the other graph, and the symmetry points of all points on the other graph are on this graph; A centrally symmetric figure means that the figure itself is centrally symmetric. The symmetry points of all points on a central symmetric graph about the center of symmetry are on the graph itself. If two centrally symmetric figures are regarded as a whole (a figure), then this figure is a centrally symmetric figure; A figure with a symmetrical center, if the symmetrical parts are regarded as two figures, then they are symmetrical about the center. That is to say:
① Centrally symmetric figure: if a figure can overlap itself after rotating around a certain point by 180 degrees, it is a centrally symmetric figure.
② Central symmetry: If one graph can overlap with another graph after rotating around a certain point by 180 degrees, then the two graphs form central symmetry. Centrally symmetric figure
Regular (2N) polygon (n is a positive integer greater than 1), line segment, rectangle, diamond, circle and parallelogram.
In fact, except for straight lines, all centrally symmetric figures have only one symmetrical point. It's just a regular figure with a central symmetrical figure.
Of course there is. It's just that the figure with central symmetry needs to meet the requirement that it is not axisymmetric. Parallelogram is the only example. It is neither an axisymmetric figure nor a centrally symmetric figure, such as an isosceles triangle and a right-angled trapezoid.
The quadratic function y = ax 2+bx+c (a, b, c are constants, and a is not equal to 0).
A>0 is opening.
A<0 opens downward.
A and B have the same sign, and the symmetry axis is on the left side of the Y axis, and vice versa.
|x 1-x2|= b 2-4ac under the radical sign divided by |a|
The intersection with the y axis is (0, c)
b^2-4ac>； 0, ax 2+bx+c = 0 has two unequal real roots.
b^2-4ac<； 0, ax 2+bx+c = 0 has no real root.
B 2-4ac = 0, AX 2+BX+C = 0 has two equal real roots.
Axis of symmetry x=-b/2a
Vertex (-b/2a, (4ac-b 2)/4a)
Vertex y = a (x+b/2a) 2+(4ac-b 2)/4a.
Function moves to the left by d (d > 0) units, the analytical formula is y = a (x+b/2a+d) 2+(4ac-b 2)/4a, and the right side is negative.
Function moves up d (d >; 0) units, the analytical formula is y = a (x+b/2a) 2+(4ac-b 2)/4a+d, and the downward direction is negative.
When a > 0, the opening is upward, the parabola is above the Y axis (the vertex is on the X axis), and it extends infinitely upward; When a < 0, the opening is downward, the parabola is below the X axis (the vertex is on the X axis), and it extends infinitely downward. The bigger the | a |, the smaller the opening; The smaller the | a |, the larger the opening.
4. When drawing a parabola y = ax2, list first, then trace points, and finally connect lines. When choosing the value of independent variable x in the list, it is often centered on 0, so choose an integer value that is convenient for calculation and tracking. When tracking points, be sure to connect them with smooth curves and pay attention to the changing trend.
Several forms of quadratic resolution function
(1) general formula: Y = AX2+BX+C (A, b, c are constants, a≠0).
(2) Vertex: y = a (x-h) 2+k (a, h, k are constants, a≠0).
(3) two expressions: y = a (X-X 1) (X-X2), where X 1, X2 is the abscissa of the intersection of parabola and x axis, that is, the two roots of quadratic equation AX2+BX+C = 0, a≠0.
Description: (1) Any quadratic function can be transformed into vertex Y = A (X-H) 2+K by formula, and the vertex coordinate of parabola is (h, k). When H = 0, the vertex of parabola Y = AX2+K is on the Y axis; When k = 0, the vertex of parabola a(x-h)2 is on the X axis; When H = 0 and K = 0, the vertex of parabola Y = AX2 is at the origin.
(2) When the parabola y = ax2+bx+c intersects the X axis, that is, the quadratic equation ax2+bx+c = 0 has the sum of the real root x 1.
When x2 exists, the quadratic function y = ax2+bx+c can be transformed into two formulas y = a (x-x 1) (x-x2) according to the decomposition formula of quadratic trinomial.
Solution of Parabolic Vertex, Symmetry Axis and Maximum Value
① collocation method: the analytical formula is transformed into the form of y = a (x-h) 2+k, with vertex coordinates (h, k) and symmetry axis as straight lines x = h, if a > 0, y has a minimum value, when x = h, y has a minimum value = k, if a < 0, y has a maximum value, and when x = h, y has a maximum value.
② Formula method: directly use the vertex coordinate formula (-,) to find its vertex; The symmetry axis is a straight line x =-, and if a > 0, y has a minimum value; When x =-, y has a minimum value =; If a < 0, y has the maximum value; When x =-, y has the maximum value =.
6. Draw an image with quadratic function Y = AX2+BX+C.
Because the image of quadratic function is parabolic and axisymmetrical, simplified point tracing method and five-point method are often used in drawing, and the steps are as follows:
(1) First, find the vertex coordinates and draw the symmetry axis;
(2) Find four points on the parabola about the axis of symmetry (such as the intersection with the coordinate axis, etc.). );
(3) Connect these five points from left to right with a smooth curve.