1. Parabola is an axisymmetric figure. The symmetry axis is a straight line x = -b/2a.
The intersection of symmetry axis and parabola is the vertex p of parabola.
Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).
pinnacle
2. A parabola has a vertex p with coordinates P (-b/2a, 4ac-B2；; )/4a)
-b/2a=0, p is on the y axis; When δ δδ= b^2; When -4ac=0, p is on the x axis.
open one's mouth
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.
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 if the axis of symmetry is on the left, the axis of symmetry is less than 0, which is -b/2a.
When a and b have different numbers (i.e. ab2a >；; 0, so b/2a should be less than 0, so a and b should have different signs.
It can be simply recorded as left and right differences, that is, when the numbers of A and B are the same (that is, AB >；; 0), the symmetry axis is on the left of the y axis; When a and b have different numbers (i.e. AB
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.
Factors determining the intersection of parabola and y axis
5. The constant term c determines the intersection of parabola and Y axis.
The parabola intersects the Y axis at (0, c)
Number of intersections between parabola and x axis
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).
When a>0, the function obtains the minimum value f(-b/2a)=4ac-b at x= -b/2a? /4a； At {x | x
{x | x >-b/2a} is an increasing function; The opening of parabola is upward; The range of the function is {y | y ≥ 4ac-b 2/4a}, and vice versa.
When b=0, the axis of symmetry of parabola is the Y axis. At this point, the function is even, and the analytical expression is transformed into y = ax 2+c (a ≠ 0).
Special value form
7. Special value form
① y=a+b+c when x =1.
② y=a-b+c when x =-1.
③ y=4a+2b+c when ③x = 2.
④ y=4a-2b+c when x =-2.
Properties of quadratic function
8. 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 whether 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 even function when b≠0.
Periodicity: None
Analytical formula:
①y = ax2+bx+c[ general formula]
⑴a≠0
⑵a & gt； 0, parabolic opening is upward; A<0, parabolic opening downward;
⑶ Extreme point: (-b/2a, (4ac-b2)/4a);
⑷δ=b^2-4ac，
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 one point:
(-b/2a，0)；
δ& lt； 0, the image does not intersect with the x axis;
②y = a(x-h)2+k[ vertex]
At this time, the corresponding extreme point is (h, k), where h=-b/2a and k = (4ac-b2)/4a;
③y=a(x-x 1)(x-x2)[ intersection (dichotomy) ](a≠0)
Axis of symmetry X=(X 1+X2)/2 when a >: 0 and X≦(X 1+X2)/2, y increases with the increase of x, when a >: 0 and x ≦ (x1+x2)/.
Decrease with the increase of …
At this time, 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 (generally connected by a quadratic equation with one variable).
Use).
The intersection point is Y=A(X-X 1)(X-X2). 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.
26.2 From the perspective of function, the quadratic equation of one variable is viewed.
1. If the parabola and the X axis have a common point, and the abscissa of the common point is 0, then the function value is 0, so it is a root of the equation.
2. There are three relationships between the image of quadratic function and X axis: there is no common point, there is a common point and there are two common points. This corresponds to three cases of roots of a quadratic equation with one variable: there are no real roots, two equal real roots and two unequal real roots.
26.3 Practical Problems and Quadratic Functions
In daily life, production and scientific research, some problems such as material saving, time saving and high efficiency can be summed up as finding the value or minimum of quadratic function.