Y = AX 2+BX+C (A ≠ 0, A, B and C are constants), then Y is called the quadratic function of X?
Important concepts: (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 quadratic. ?
X is an independent variable and y is a quadratic function of X?
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Three expressions of quadratic function?
① General formula: y = ax 2; +bx+c(a, b, c are constants, a≠0)?
② Vertex [Vertex of parabola? P(h，k)？ ]:y=a(x-h)^2+k？
(3) Intersection [only intersects with the X axis? A(x 1，0)？ And then what? B(x2，0)？ Parabola of ]: y = a (x-x 1) (x-x2)?
The above three forms can be transformed as follows:?
What is the relationship between (1) general formula and vertex?
For the quadratic function y=ax+bx+c, its vertex coordinates are (-b/2a), (4ac-b2)/4a), that is?
h=-b/2a=(x 1+x2)/2？
k =(4ac-b & amp； sup2)/4a？
② What is the relationship between the general formula and the intersection point?
x 1，x2 =[-b √( b & amp； Sup2-4ac)]/2a (that is, the formula for finding the root of a quadratic equation with one variable)?
{Generally, the intersection point is used less, because it is better to find it directly with the general formula when there are enough points}?
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Let the quadratic function y = x&in the plane rectangular coordinate system; The image of sup2?
It can be seen that the image of quadratic function is an endless parabola. ?
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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 parabola is Y axis (that is, straight line x=0)?
2. A parabola has a vertex p with coordinates p? (? -b/2a？ ，(4ac-b & amp； sup2)/4a？ )?
-b/2a=0, p is on the y axis; When δ =? B & ampsup2 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 upward; 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 the signs of A and B are the same (that is, AB > 0), the symmetry axis is left on the 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 the signs of A and B are different (that is, AB < 0), the symmetry axis is on the right side of the Y axis. Because the axis of symmetry is on the right, it is greater than 0, that is,-b/2a >; 0, so b/2a should be less than 0, so a and b should have different symbols?
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. ?
5. The constant term c determines the intersection of parabola and Y axis. ?
The parabola intersects the Y axis at (0, c)?
6. What is the number of intersections between parabola and X axis?
Δ=? B2； When -4ac > 0, the parabola has two intersections with the x-axis. ?
Δ=? B2； When -4ac=0, the parabola has 1 intersections with the X axis. ?
Δ=? B2； When -4ac < 0, the parabola has no intersection with the x axis. The value of x is imaginary (x=? -b√B2； -4ac？ The reciprocal of the value of, multiplied by the imaginary number I, the whole formula divided by 2a)?
When a>0, the function is at x=? The minimum value f (-b/2a) = 4ac-b&; sup2/4a； In {x | x-b/2a} is an increasing function; The opening of parabola is upward; The range of the function is {y | y ≥ 4ac-b2; /4a} The reverse side remains the same?
When b=0, the axis of symmetry of parabola is the Y axis. At this point, the function is an even function, and the analytic expression is transformed into y=ax2. +c(a≠0)？
{Remember the formula of symmetry axis: the left and right are different. (The symmetry axis is on the left of the Y axis, and the symbols of A and B are the same. The symmetry axis is on the right side of the Y axis, and A and B are different symbols)