Chapter 26 Quadratic Function
26. 1 quadratic function and its image
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.
Generally speaking, there is the following relationship between independent variable x and dependent variable y:
general formula
y = ax∧2； +bx+c(a≠0, a, b and c are constants), and the vertex coordinates are (-b/2a,-4ac-b ∧ 2)/4a);
Vertex type
Y=a(x+m)∧2+k(a≠0, a, m and k are constants) or y=a(x-h)∧2+k(a≠0, a, h and k are constants), and the vertex coordinates are (-).
Intersection formula
Y = A(X-X 1)(X-x2)[ only applicable to parabolas where A (X 1 0) and B (X2, 0) intersect with the x axis];
Important concepts: a, b and c are constants, a≠0, a determines the opening direction of the function, a >；; 0, the opening direction is upward, a
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))。 Therefore, the intersection coefficient a = y1/(x1* x2) (y1is the intercept) can be derived.
Root formula
The right side of a quadratic function expression is usually a quadratic trinomial.
Root formula
X is an independent variable and y is a quadratic function of X.
x 1，x2=[-b (√(b^2-4ac))]/2a
(that is, the formula for finding the root of a quadratic equation with one variable) (as shown on the right)
There are factorization method and collocation method to find the root.
Make an image of the square of the quadratic function y=2x in the plane rectangular coordinate system,
It can be seen that the image of quadratic function is an endless parabola.
Different quadratic function images
If the drawn figure is accurate, then the quadratic function will be translated by the general formula.
Note: The sketch itself should have an image of 1, and the function should be indicated next to it.
Draw the symmetry axis and point out that X=
3 coordinates intersecting with X axis, coordinates intersecting with Y axis, and coordinates of vertices. Properties of parabola
Axial symmetry
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).
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. ab; 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+ when x =-1.