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Quadratic function refers to a polynomial function whose highest unknown degree is quadratic. The quadratic function can be expressed as f (x) = ax 2+bx+c (a is not 0). Its image is a parabola, and its principal axis is parallel to the Y axis.

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Define and define the solution of expression quadratic function

general formula

Vertex type

Intersection formula

Newton interpolation formula (finding resolution function at three known points)

Root formula

How to learn quadratic function

Image Axisymmetry of Quadratic Function

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Factors determining the position of symmetry axis

Factors determining the intersection of quadratic function image and y axis

Number of intersection points between quadratic function image and x axis

Special value form

Properties of quadratic function

Symmetry of two images

Definition and definition expression of quadratic function and quadratic equation of one variable The solution of quadratic function

general formula

Vertex type

Intersection formula

Newton interpolation formula (finding resolution function at three known points)

Root formula

How to learn quadratic function

Image Axisymmetry of Quadratic Function

pinnacle

open one's mouth

Factors determining the position of symmetry axis

Factors determining the intersection of quadratic function image and y axis

Number of intersection points between quadratic function image and x axis

Special value form

Properties of quadratic function

Symmetry of two images

Quadratic function and unary quadratic equation

Expand quadratic function and its image

Edit this paragraph definition and definition expression.

Generally speaking, there is the following relationship between the independent variable (usually X) and the dependent variable (usually Y):

Solution of quadratic function

The general formula of quadratic function is that y is equal to a times x plus the square of b times x plus c, and written by mathematical equation is y = ax+bx+c. If you know three points, you can bring in the coordinates of the three points, which means that three equations can solve three unknowns, such as equation 1 8=a? ^+b？ +c Simplifies 8=c, which means that c is the intersection of the function and the y axis. Equation 2 7 = A * 6 2+B * 6+C Simplification 7=36a+6b+c Equation 3 7 = A *(6)2+B *(6)+C Simplification 7=36a-6b+c Solve abc, and the above one is old. 7) These two coordinates can be used to find an axis of symmetry, that is, X=0. It can also be calculated by the symmetry axis formula x=-b/2a. If we know the two coordinates of the X axis (the values of the two coordinates of y=0 are called the two roots of this equation), we can also use the symmetry axis formula x=-b/2a or the unary quadratic equation ax+bx+c=0 (a≠0 and△ =

general formula

Y=ax+bx+c(a≠0, a, b and c are constants), and the vertex coordinates are (-b/2a, 4ac-b? /4a)；

Vertex type

y=a(x-h)？ ; +k(a≠0, a, h, k are constants), vertex coordinates are (h, k), symmetry axis is x=h, and the position characteristics of vertices and the opening direction of images are related to the function y=ax? ; The images are the same, and sometimes the topic will point out that the general formula can be turned into a vertex by collocation;

Intersection formula

Y = a(X-X 1)(X-x2)(a≠0)[ only applicable to parabola intersecting with X axis, that is, y=0, that is, B2-4ac ≥ 0]; Steps to change from general formula to intersection: ∵ x1+x2 =-b/ax1x2 = c/a ∴ y = ax? ; +bx+c=a(x？ ; +b/ax+c/a)=a[﹙x？ ; -(x1+x2) x+x1x2] = a (x-x1) (x-x2) Important concepts: a, b and c are constants, and a≠0, a determines the opening direction of the function. A>0, the opening direction is upward; A<0, the opening direction is downward. The absolute value of a can determine the opening size. The greater the absolute value of a, the smaller the opening, and the smaller the absolute value of a, the larger the opening.

Newton interpolation formula (finding resolution function at three known points)

y =(y3(x-x 1)(x-x2))/((x3-x 1)(x3-x2)+(y2(x-x 1)(x-x3))/((x2-x 1)(x2-x3)+(y 1(x-x2)(x-x3))/((x 1-x2)(x 1-x3))。 From this, the formula for finding the root of the intersection coefficient A = y1/(x1x2) (y1is the intercept) can be derived.

The right side of a quadratic function expression is usually a quadratic trinomial.

Root formula

X is an independent variable, y is a quadratic function of x x 1, and x2 = [-b (√ (b? ; -4ac)]/2a (that is, the formula for finding the root of a quadratic equation with one variable) (as shown on the right) The factorization method and collocation method for finding the root. When the intersection of quadratic function and x axis is △=b? ; -4ac & gt； 0, the function image has two intersections with the x axis. When △=b? ; When -4ac=0, the function image intersects the x axis. When △=b? ; -4ac & lt； 0, the function image does not intersect with the x axis.

How to learn quadratic function by editing this paragraph?

1。 Understand the meaning of the function. 2。 Remember several expressions of the function and pay attention to the differences. 3。 General formula, vertex, intersection, etc. , distinguish the difference between symmetry axis, vertex, image, etc. 4。 Understanding function images with practice. 5。 When calculating, remember to take the value range when looking at the image.

Edit the image of the quadratic function in this paragraph.

Make the image of quadratic function y=ax2+bx+c in the plane rectangular coordinate system. It can be seen that the image of quadratic function is an endless parabola. If the drawn graph is accurate, then the quadratic function image will be obtained by general translation. Note: The sketch itself should have an image of 1, and the function should be indicated next to it. 2. Draw the symmetry axis and point out what the straight line X is (X=-b/2a) 3. Intersection point of coordinate straight line X and X axis (x 1, y1); (x2, y2), the coordinate (0, c) intersecting with the Y axis, and the coordinate of the vertex (-b/2a, (4ac-bx? )/4a)。 Properties of parabola

Axial symmetry

The image of 1. quadratic function is axisymmetric. The symmetry axis is a straight line x = h or x=-b/2a, and the only intersection point between the symmetry axis and the quadratic function image is the vertex p of the quadratic function image. Especially when h=0, the symmetry axis of quadratic function image is Y axis (that is, straight line x=0) a and B with the same sign, the symmetry axis is on the left side of Y axis b=0, the symmetry axis is Y axis A and B with different signs, and the symmetry axis is on the right side of Y axis.

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2. The image of quadratic function has a vertex p whose coordinate is P (h, k). When h=0, p is on the y axis; When k=0, p is on the x axis. h=-b/2a k=(4ac-b2)/4a

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3. The quadratic coefficient A determines the opening direction and size of the quadratic function image. When a>0, the quadratic function image opens upward; When a<0, the parabola opens downward. The larger the |a|, the smaller the opening of the quadratic function image.

Factors determining the position of symmetry axis

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; Because the axis of symmetry is on the left, the axis of symmetry is less than 0, which is -b/2a.

Factors determining the intersection of quadratic function image and y axis

5. The constant term c determines the intersection point between the quadratic function image and the Y axis. The quadratic function image intersects the Y axis at (0, c). Note: The coordinate of the vertex is (h, k), and it intersects with the Y axis at (0, c).

Number of intersection points between quadratic function image and x axis

6. The number of intersections between the quadratic function image and the X axis is A: 0, and the function gets the minimum value ymix=k at x=h, and at X; K when a < 0, the function obtains the maximum value ymax=k at x=h and at x >; Increasing function in the range of h, in x

Special value form

7. The form of special value ①x =- 1y = A+AH2+2AH+K2 when Y = A+AH2-2ah+K3Y = 4A+AH2+8ah+K4X =-2Y = 4A+AH.

Properties of quadratic function

8. Definition domain: R value domain: (corresponding to the analytical formula, and only discuss 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 2)/4a, positive infinity); ②[t, positive infinity] parity: an even function when b=0 and an odd non-even function when b≠0. Periodicity: no analytical formula: ①y=ax? +bx+c [general formula] (1) A ≠ 0 (2) A > 0, and the parabolic opening is upward; A<0, parabolic opening downward; (3) Extreme point: (-b/2a, (4ac-b? ; )/4a)； ⑸δ= B2-4ac，δ& gt； 0, 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 a point: (-b/2a, 0); δ& lt； 0, the image does not intersect with the x axis; ②y=a(x-h)2+k[ vertex] The corresponding extreme point is (h, k), where h=-b/2a and k=(4ac-b? )/4a； ③y=a(x-x 1)(x-x2)[ intersection point (dichotomy) ](a≠0) Symmetry axis X=(X 1+X2)/2 when a >: 0 and x ≧ x (x/kloc-). Know the intersection of two x axes and the coordinates of another point. The x value of two intersection points is the corresponding X 1 X2 value.

Symmetry of two images

①y=ax2+bx+c and y=ax2-bx+c are symmetric about y; ②y=ax2+bx+c, y=-ax2-bx-c is symmetrical about x; ③y=ax2+bx+c and y=-a(x-h﹚2+k is symmetric about the vertex; ④y=ax2+bx+c and y=-a(x+h﹚2-k is symmetrical about the origin.

Edit the quadratic function and quadratic equation of this paragraph.

Especially quadratic function (hereinafter referred to as function) y=ax? +bx+c, when y=0, the quadratic function is a unary quadratic equation about x (hereinafter referred to as the equation), that is, ax? +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? ; ; ; ，y=a(x-h)？ ; ; ; ，y=a(x-h)？ ; ; +k，y=ax？ ; ; The images of +bx+c (various a≠0) have the same shape but different positions. Their vertex coordinates and symmetry axes are as follows: analytic vertex coordinates symmetry axis y=ax? (0，0) x=0 y=ax？ ; +K (0，K) x=0 y=a(x-h)？ ; (h，0) x=h y=a(x-h)？ ; +k (h，k) x=h y=ax？ ; +bx+c (-b/2a，4ac-b？ ; /4a) x=-b/2a when h >; 0，y=a(x-h)？ ; ; Can the image be represented by parabola y=ax? ; ; By moving H units in parallel to the right, when H; 0; When a<0, the image falls below the X axis, and when X is an arbitrary real number, there is Y; 0(a & lt； 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 analytical formula of quadratic function (1) by the method of undetermined coefficient. When the given condition is that the known image passes through three known points or three pairs of known corresponding values of x and y, the analytical formula can be set to the general form: y=ax? ; +bx+c(a≠0).(2) When the given condition is the vertex coordinate or symmetry axis or the maximum (minimum) value of the known image, the analytical formula can be set as the vertex: y=a(x-h)? ; +k(a≠0).(3) When the given condition is that the coordinates of the two intersections of the image and the X axis are known, the analytical formula can be set as two formulas: y=a(x-x 1)(x-x2)(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. A typical example of the senior high school entrance examination is 1. (Dongcheng District, Beijing) has an image of a quadratic function. Three students described some characteristics of it: a: the symmetry axis is a straight line x = 4；; B: the abscissa of the two intersections with the X axis is an integer; C: The ordinate intersecting with the Y axis is also an integer, and the area of the triangle with these three intersections as its vertices is 3. Please write a quadratic resolution function that satisfies all the above characteristics. Test site: quadratic function y=ax? ; Comment on the solution of +bx+c: let the analytical formula be y=a(x-x 1)(x-x2) and x 1