Knowledge point of linear function 1. linear function
If y=kx+b(k and b are constants, k≠0), then y is called a linear function of x.
Especially when b=0, the linear function y=kx+b becomes y=kx(k is constant, k≠0), and y is called the proportional function of X.
2. Images and properties of linear functions
Any point P(x, y) on the (1) linear function satisfies the equation: y = kx+b.
(2) The coordinate of the intersection of the linear function and the Y axis is always (0, b), and the coordinate of the intersection of the linear function and the X axis is always (-b/k, 0).
(3) The image of the proportional function always passes through the origin.
(4) The relationship between k, b and the quadrant where the function image is located:
When k>0, y increases with the increase of x; When k < 0, y decreases with the increase of x.
When k>0, b>0, the straight line passes through the first, second and third quadrants;
When k>0, b<0, the straight line passes through the first, third and fourth quadrants;
When k < 0, b>0, a straight line passes through the first, second and fourth quadrants;
When k < 0, b<0, a straight line passes through the second, third and fourth quadrants;
When b=0, the straight line passing through the origin o (0 0,0) represents the image of the proportional function.
At this time, when k>0, the straight line only passes through the first and third quadrants; When k < 0, the straight line only passes through the second and fourth quadrants.
Knowledge point of quadratic function 1. Quadratic function expression
(1) vertex type
y=a(x-h)？ +k(a≠0, a, h, k are constants), the vertex coordinates are (h, k), the symmetry axis is a straight line x=h, and the position characteristics of the vertex and the opening direction of the image are related to the function y=ax? When x=h, y = the maximum (minimum) value of k.
(2) Intersection point
y=a(x-x？ )(x-x？ ) [limited to the parabola intersecting with the x axis, that is, y=0, that is, b? -4ac & gt； 0]
Function and image intersect at (x? 0) and (x? ,0)
(3) general formula
y=aX？ +bX+c=0(a≠0)(a, B and C are constants)
2. Symmetry axis of quadratic function
Quadratic function image is an axisymmetric figure. The symmetry axis is a straight line x=-b/2a.
The only intersection of the symmetry axis and the quadratic function image is the vertex p of the quadratic function image.
Especially when b=0, the symmetry axis of the quadratic function image is the Y axis (that is, the straight line x=0).
A and B have the same sign, and the symmetry axis is on the left side of Y axis;
A and B are different symbols, and the symmetry axis is on the right side of the Y axis.
3. Symmetry relation of quadratic function image
(1) For the general formula:
①y=ax2+bx+c and y=ax2-bx+c are symmetric about Y ..
② two images y = ax2+bx+c and y=-ax2-bx-c are symmetrical about X ..
③y=ax2+bx+c and y=-ax2-bx+c-b2/2a are symmetrical about the vertex.
④y=ax2+bx+c and y=-ax2+bx-c are symmetrical about the center of the origin. (i.e., the graph obtained after rotating 180 degrees around the origin)
(2) For vertices:
① two images, y = a (x -h) 2+k and y=a(x+h)2+k, are symmetrical about y axis, that is, vertices (h, k) and (-h, k) are symmetrical about y axis, and the abscissas are opposite, but the ordinate is the same.
② two images, y = a (x-h) 2+k and y=-a(x-h)2-k, are axisymmetrical about x, that is, vertices (h, k) and (h, -k) are axisymmetrical about x, with the same abscissa and opposite ordinate.
③y=a(x-h)2+k and y=-a(x-h)2+k are symmetrical about the vertices, that is, the vertices (h, k) and (h, k) are the same and the opening directions are opposite.
④y=a(x-h)2+k and y=-a(x+h)2-k are symmetrical about the origin, that is, the vertices (h, k) and (-h, -k) are symmetrical about the origin, and the abscissa and ordinate are opposite.