The simplest and most common function, the image is a straight line in the plane rectangular coordinate system.

Domains (unless otherwise specified below, they are all domains with no special requirements): r

Scope: r

Parity: None

Periodicity: None

Analytical formula of plane rectangular coordinate system (hereinafter referred to as analytical formula);

①ax+by+c=0 [general formula]

②y=kx+b[ oblique]

(k is the slope of the straight line, b is the longitudinal intercept of the straight line, and the proportional function b=0).

③y-y 1=k(x-x 1)[ point inclination]

(k is the slope of the straight line, (x 1, y 1) is the point where the straight line passes)

④ (y-y1)/(y2-y1) = (x-x1)/(x2-x1) [two-point formula]

((x 1, y 1) and (x2, y2) are two points on a straight line)

⑤x/a-y/b=0[ intercept type]

(A and B are the intercepts of a straight line on the X axis and the Y axis, respectively)

Limitations of analytical expressions:

① More requirements (3);

② and ③ cannot express straight lines without slope (straight lines parallel to the X axis);

④ There are many parameters and the calculation is too complicated;

⑤ Cannot represent a straight line parallel to the coordinate axis and a straight line passing through a point.

Inclination angle: The included angle between the X axis and the straight line (the angle formed by the straight line and the positive direction of the X axis) is called the inclination angle of the straight line. Let the inclination of the straight line be a, and the slope of the straight line is k=tg(a).

2. Quadratic function

The common function in the topic is that the image in the plane rectangular coordinate system is a parabola whose symmetry axis is parallel to the Y axis.

Domain: r

Range: ① [(4ac-b 2)/4a, positive infinity]; ②[t, positive infinity]

Parity: even function

Periodicity: None

Analytical formula:

①y = ax2+bx+c[ general formula]

⑴a≠0

(2) when a > 0, the parabolic opening is upward; A < 0, parabolic opening downward;

⑶ Extreme point: (-b/2a, (4ac-b2)/4a);

⑷δ=b^2-4ac，

δ> 0, where 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 < 0, the image has no intersection with the X axis;

②y = a(x-h)2+t[ collocation]

The corresponding extreme point is (h, t), where h=-b/2a and t = (4ac-b2)/4a);

3. Inverse proportional function

The image in the plane rectangular coordinate system is a hyperbola.

Domain: (negative infinity, 0)∩(0, positive infinity)

Range: (negative infinity, 0)∩(0, positive infinity)

Parity check: odd function

Periodicity: None

Analytical formula: y= 1/x

4. Power function

y=x^a

①y=x^3

Domain: r

Scope: r

Parity check: odd function

Periodicity: None

This image is similar to passing the fourth interval part of the quadratic function through a point symmetrical about the X axis.

Post-image (analogy, this method can't get cubic function image)

②y=x^( 1/2)

Domain: [0, positive infinity]

Range: [0, positive infinity]

Parity: None (that is, neither odd nor even)

Periodicity: None

The image is similar to a quadratic function that rotates clockwise through a point with the origin as the rotation center.

90, and then remove the part below the y axis to get the image (analogy, this method can't be obtained three times.

Functional image)

5. Exponential function

An image in a plane rectangular coordinate system (it's too difficult to describe, let's talk about its properties first ...)

Constant intersection (0, 1). According to the analytical formula, if a > 1, the function monotonically increases in the definition domain; If 0 < a < 1, the function is simply reduced in the domain.

Domain: r

Range: (0, positive infinity)

Parity: None

Periodicity: None

Analytical formula: y = a x

a>0

Property: It is the reciprocal of logarithmic function y = log (a) x.

* logarithmic expression: log(a)x represents the logarithm with x as the base.

6. Logarithmic function

The image on the definition domain is symmetrical with the image of the corresponding exponential function (inverse function of logarithmic function) about the straight line y = X.

Constant crosses the fixed point (1, 0). According to the analytical formula, if a > 1, the function monotonically increases in the definition domain; If 0 < a < 1, the function is simply reduced in the domain.

Domain: (0, positive infinity)

Scope: r

Parity: None

Periodicity: None

Analytical formula: y=log(a)x

a>0

Property: it is the reciprocal of logarithmic function y = a x.

7. trigonometric functions

① Sine function: y=sinx

The image is a sine curve (a wavy line, which is the basis of all curves).

Domain: r

Range: [- 1, 1]

Parity check: odd function

Periodicity: The minimum positive period is 2π.

Symmetry axis: straight line x=kπ/2 (k∈Z)

Center symmetry point: intersection point with X axis: (kπ, 0)(k∈Z)

⑵ Cosine function: y=cosx

The image is a sine curve, which is obtained by shifting the image of sine function to the left by π/2 units (minimum shift).

Domain: r

Range: [- 1, 1]

Parity: even function

Periodicity: The minimum positive period is 2π.

Symmetry axis: straight line x=kπ (k∈Z)

Central symmetry point: intersection point with X axis: (π/2+kπ, 0)(k∈Z)

⑶ tangent function: y=tg x

Each periodic unit of an image is like a cubic function, in which many periodic units are evenly distributed on the X axis.

Domain: {x│x≠π/2+kπ}

Scope: r

Parity check: odd function

Periodicity: The minimum positive period is π

Symmetry axis: none

Center symmetry point: intersection point with X axis: (kπ, 0)(k∈Z).

* The properties of trigonometric functions are slightly more, and there are more than one thousand light formulas. In addition, the image translation and stretching changes of trigonometric functions have been clearly explained in the content of image translation (not here, but in the textbook), so I won't say much.

You're done! I hope it will be helpful to your study.