Independent variable x and dependent variable y have the following relationship:
Y=kx+b(k, b is a constant, k≠0)
It is said that y is a linear function of x.
In particular, when b=0, y is a proportional function of x.
Two. Properties of linear functions:
The change value of y is directly proportional to the corresponding change value of x, and the ratio is K.
That is △ y/△ x = K.
Three. Images and properties of linear functions;
1. exercise and graphics: through the following three steps list (1) (generally find 4-6 points); (2) tracking points; (3) Connection, you can make an image of a function. (Connected by a smooth straight line)
2. Property: any point P(x, y) on the image of a linear function satisfies the equation: y = kx+b.
3. Quadrant where k, b and function images are located.
When k > 0, the straight line must pass through the first and third quadrants, and y increases with the increase of x;
When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.
When b > 0, the straight line must pass through the first and second quadrants; When b < 0, the straight line must pass through three or four quadrants.
Particularly, when b=0, the image of the proportional function is represented by a straight line of the origin o (0 0,0).
At this time, when k > 0, the straight line only passes through one or three quadrants; When k < 0, the straight line only passes through two or four quadrants.
Four. Determine the expression of linear function:
Known point A(x 1, y1); B(x2, y2), please determine the expressions of linear functions passing through points A and B. ..
(1) Let the expression (also called analytic expression) of a linear function be y = kx+b.
(2) Because any point P(x, y) on the linear function satisfies the equation y = kx+b, two equations can be listed:
Y 1 = KX 1+B 1，Y2 = KX2+B2。
(3) Solve this binary linear equation and get the values of K and B. ..
(4) Finally, the expression of the linear function is obtained.
V, in y=kx+b, two coordinate systems must pass through (0, b) and (-b/k, 0).
Application of linear function of intransitive verbs in life
1. When the time t is constant, the distance s is a linear function of the velocity v .. s=vt.
2. When the pumping speed f of the pool is constant, the water quantity g in the pool is a linear function of the pumping time t, and the original water quantity s in the pool is set. G = S- feet.
inverse proportion function
A function in the form of y = k/x (where k is a constant and k≠0) is called an inverse proportional function.
The range of the independent variable x is all real numbers that are not equal to 0.
The image of the inverse proportional function is a hyperbola.
As shown in the figure, the function images when k is positive and negative (2 and -2) are given above.
quadratic function
Generally speaking, there is the following relationship between independent variable x and dependent variable y:
y=ax^2+bx+c (a≠0)
(a, b, c are constants, a≠0, a determines the opening direction of the function, a >；; 0, the opening direction is upward, a
Y is called the quadratic function of X.
The right side of a quadratic function expression is usually a quadratic trinomial.
X is an independent variable and y is a function of X.
Three Expressions of Quadratic Function
General formula: y = ax 2+bx+c (a, b and c are constants, and a≠0).
Vertex: y = a(x-h)2+k[ vertex P(h, k)] For quadratic function y = ax 2+bx+c, its vertex coordinates are (-b/2a, (4ac-b 2)/(4a)).
Intersection point: y=a(x-x? 6? 9)(x-x？ 6? 0) 【 only when it is related to the x-axis A(x? 6? 9,0) and B(x? 6? 0,0) parabola]
Where x 1, 2 = (-b √ (b 2-4ac))/(2a)
Note: Among these three forms of mutual transformation, there are the following relations:
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h=-b/(2a) k=(4ac-b^2)/(4a) x？ 6? 9，x？ 6? 0=(-b √b^2-4ac)/2a
Quadratic function image
The image of quadratic function y = x 2 in plane rectangular coordinate system,
Quadratic function As you can see, the image of quadratic function is a parabola.
Standard drawing steps of quadratic function
(on a plane rectangular coordinate system)
(1) list
(2) Tracking point
(3) Connection
Properties of parabola
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).
2. The parabola has a vertex p, and the coordinate is P (-b/2a, (4ac-b 2)/4a).
-b/2a=0, p is on the y axis; When δ = b 2-4ac = 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;
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.
5. The constant term c determines the intersection of parabola and Y axis.
The parabola intersects the Y axis at (0, c)
6. Number of intersections between parabola and X axis
When δ = b 2-4ac > 0, the parabola has two intersections with the X axis.
When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.
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When 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-b2/4a at x= -b/2a; In {x | x-b/2a} is an increasing function; The opening of parabola is upward; The range of the function is {x | x ≥ 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).
Quadratic function and unary quadratic equation
In particular, the quadratic function (hereinafter called function) y = ax 2+bx+c,
When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).
That is, ax 2+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 2, Y = A (X-H) 2, Y = A (X-H) 2+K, y = ax 2+bx+c (among all kinds, a≠0) has the same image shape, but different positions.
Analytical formula
y=ax^2
y=a(x-h)^2
y=a(x-h)^2+k
y=ax^2+bx+c
Vertex coordinates
(0,0)
(h，0)
(h，k)
(-b/2a,(4ac-b^2)/4a)
axis of symmetry
x=0
x=h
x=h
x=-b/2a
When h>0, the parabola y = ax 2 is moved to the right by H units in parallel, and the image of y = a (x-h) 2 can be obtained.
When h < 0, it is obtained by moving |h| units in parallel to the left.
When h>0, k>0, the parabola y = ax 2 is moved to the right by H units in parallel, and then moved up by K units, the image of y = a (x-h) 2+k can be obtained;
When h>0, k<0, the parabola y = ax 2 is moved to the right by h units in parallel, and then moved down by | k units, and the image of y = a (x-h) 2+k is obtained;
When h < 0, k >; 0, moving the parabola to the left by |h| units in parallel, and then moving it up by k units to obtain an image with y = a (x-h) 2+k;
When h < 0, k<0, move the parabola to the left by |h| units in parallel, and then move it down by |k| units to obtain an image with y = a (x-h) 2+k;
Therefore, it is very clear to study the image of parabola y = ax 2+bx+c (a ≠ 0) and change the general formula into the form of Y = A (X-H) 2+K through the formula, so as to determine its vertex coordinates, symmetry axis and approximate position of parabola, which provides convenience for drawing images.
2. the image of parabola y = ax 2+bx+c (a ≠ 0): when a >: 0, the opening is upward, when a.
3. parabola y = ax 2+bx+c (a ≠ 0), if a >；; 0, when x ≤ -b/2a, y decreases with the increase of x; When x ≥ -b/2a, y increases with the increase of x, if a
4. The intersection of the image with parabola y = ax 2+bx+c and the coordinate axis:
(1) The image must intersect with the Y axis, and the coordinate of the intersection point is (0, c);
(2) when △ = b 2-4ac >; 0, the image intersects the x axis at two points A(x? 6? 9,0) and B(x? 6? 0,0), where x 1, x2 is the unary quadratic equation ax 2+bx+c = 0.
(a≠0)。 The distance between these two points AB=|x? 6? 0-x？ 6? In addition, the distance between any pair of symmetrical points on the parabola can be | 2× (-b/2a)-a | (a is one of them).
When △ = 0, the image has only one intersection with the X axis;
When delta < 0. The image does not intersect with the x axis. When a >; 0, the image falls above the X axis, and when X is an arbitrary real number, there is y >；; 0; When a<0, the image falls below the X axis, and when X is an arbitrary real number, there is Y.
5. the maximum value of parabola y = ax 2+bx+c: if a>0 (a <; 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 analytic expression of quadratic function by undetermined coefficient method.
(1) When the given condition is that the known image passes through three known points or three pairs of corresponding values of known x and y, the analytical formula can be set to the general form:
y=ax^2+bx+c(a≠0).
(2) When the given condition is the vertex coordinate or symmetry axis of the known image, the analytical formula can be set as the vertex: y = a (x-h) 2+k (a ≠ 0).
(3) When the given condition is that the coordinates of two intersections between the image and the X axis are known, the analytical formula can be set as two formulas: y=a(x-x? 6? 9)(x-x？ 6? 0)(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.