Middle School Function Analysis, Middle School Math, Functions

y=sinh(x).Define the domain: r

Range: rOdd functions.

The function image is a strictly monotonically increasing curve that crosses the origin and crosses , quadrant, and is the equivalent infinity of (1 2)e x when x->+. The function image is symmetrical with respect to the origin.

y=cosh(x).Define the domain: r

Range: [1,+ even function.] The function image is a catenary with a minimum point of (0,1) and a strictly monotonic increasing curve in the quadrant section, which is the equivalent infinity of (1 2)e x when x->+.

The function image is symmetrical with respect to the y-axis.

y=tanh(x).Define the domain: r

Range: (-1,1)Odd functions.

The function image is a strictly monotonically increasing curve that crosses the origin and crosses the quadrant. The image is confined to two asymptotes y=1 and y=-1. lim[x->+tanh(x)=1],lim[x->-tanh(x)=-1].

y=coth(x).Define the domain:Range:

Odd functions. The function image is divided into two branches, respectively in the , quadrant, and the function in (- 0) and (0,+ which are monotonically decreasing. The vertical asymptote is the y-axis, and the two horizontal asymptotic lines are y=1 and y=>+coth(x)=1], and lim[x->-coth(x)=-1].

y=sech(x).Define the domain: r

Range: (0,1).Even functions.

The highest point is (0,1), and the function is strictly monotonically decreasing at (0,+. The x-axis is its asymptote. lim[x->∞sech(x)]=0.

y=csch(x).Define the domain:Range:

Odd functions. The function image is divided into two branches, respectively in the , quadrant, and the function in (- 0) and (0,+ which are monotonically decreasing. The vertical asymptote is the y-axis, and the two horizontal asymptotes are the x-axis.

lim[x->∞csch(x)]=0.

The name of the hyperbolic function has been changed: SH is also called sinh, ch is also called cosh

In a process of change, the amount of change is called a variable, and some values do not change with the variable, we call them constants.

An independent variable, a variable that is associated with another quantity, any value of which can find a fixed value in the other quantity.

The dependent variable (function) changes with the change of the independent variable, and when the independent variable takes a unique value, the dependent variable (function) has and only a unique value corresponding to it.

What type of function do you want to know??? It's hard to say

The value of the function, in a function where y is x, x determines a value, and y determines a value, and when x takes a, y is determined as b, and b is called the function value of a.

It is known that y=ax +bx+3 finds the analytic formula of the quadratic function and the coordinates of the vertex m through a(-1,0),b(3,0).

Solution: Substitute a(-1,0),b(3,0) into y=ax +bx+3.

a-b+3=0

9a+3b+3=0

The solution yields {a=-1

b=2, so the analytic formula of this quadratic function is y=-x +2x+3, and since y=-x +2x+3=-(x-1) +4, then the coordinates of the vertex m of the quadratic function are (1,4).

The coordinates of the intersection of the function y=2x-4 image with the x-axis are (2,0) and (0,-4) with the y-axis

The function is obtained by translating (2) units of the image of the function y=3x (to the right) and (4) units (to the top).

y=3（x-2）²+4

(1) When the straight line y=-x+2 and the hyperbolic y=kx -1(k≠0) have two different intersections, -x+2=kx -1 has two different roots, which is obtained by the equation -x+2=kx -1.

x^2-2x+k=0,……

x^2-2x+1=1-k

x-1) 2=1-k, so there is only 1-k>0 and k≠0, i.e., k<1 and k≠0.

2) From the equation (*), x1+x2=2, x1*x2=k, x1 2+x2 2=(x1+x2) 2-2x1*x2=4-2k

such that (x1-2)(x2-2)=x2 x1+x1 x2, i.e.

x1x2-2(x1+x2)+4=(x1^2+x2^2)/x1x2

k-2*2+4=(4-2k)/k

k^2+2k-4=0

k1=-1+root5,k2=-1-root5;

where k1>1 is not in place, k2<1

There is k=-1-root number 5 such that (x1-2)(x2-2)=x2 x1+x1 x2

It's a high school problem. I didn't learn hyperbola in junior high school.

This question is very simple to use a combination of several lines, and you draw two figures. By translating the straight line, several limit cases are found.

To graph, this problem involves an inverse proportional function and a primary function.

Solution: (Lu Qixian demolition 1) The number of type C diesel generators is 10-x-y 4x+3y+2(10-x-y)=32

y=12-2x

2) The C diesel generator is 10-x-y=(x-2) units.

w=130x+120(12-2x)+100(x-2)-10x+1240

According to the problem of the inequality, we get: 3 x

x is a positive integer x=3,4,5

w decreases with the increase of x's sideline when x=5, w is at least -10 5+1240=1190 (yuan).

Solution: (1) 1. The number of pure Qing of C diesel generator is 10-x-y.

x+y=12

2) w = 30x + 20y + 1000 Lianli 2x + y = 12 to get w = 1240-10x, because at least one of each, w up to 8 different searches. Therefore, 8 units of A, 1 unit of B, and 1 unit of C, which can pump water for irrigation as required, and at the same time, the total cost of diesel generators is the least.

If the ** of the two companies A and B is the same, the total cost of the two companies is the same.

Company A Fees:

Company B Fees:

ax (x<=10)

10a+3a (x=20, buy 20 computers, the cost of the two companies is the same, x>20, company B is cheaper, x<20, company A is cheaper.

Solution: Let the ** of company A and company B be k, the total cost of purchasing computers in company A is y1, and the total cost of purchasing computers in company B is y2, and the number of units is x

y1=y2=10k+

When y1>y2, >10k+, the solution: x>20 When y1=y2, the solution gets: x=20

When Y1: When the total number of computers purchased by the school is equal to 20, the purchase cost of the two companies is equal, and when the total number of computers purchased by the school is greater than 20, the purchase of the school in company B saves money.

Suppose you buy x computers, the unit price is A, and the total cost is Y

A: y = B: y = ax (x greater than or equal to 0, less than or equal to 10) y = 10a+

From the above, it can be seen that when x is greater than or equal to 0, less than or equal to 20, choose A, and when x is greater than 20, select B.

If the result is not very well understood, you can make two function graphs, where A is a primary function and B is a piecewise function, and the result can be easily seen. This kind of question is also mostly this method, intuitive)

Let ** be a

Ten are suitable for an inner armor company.

More than 10 units are suitable for Company B.

a*85%*10

a*70%*10

Solution: The straight line L1 is (4,1)(-2,3), and the equation of the straight line L1 is: (Y1-1) (X-4)=(3-1) (2-4), Y1=-1 Tong contains 3X+7 3

The straight line L2 passes through the point: (-1, 0) (vibrillation 0, -3), and the equation for the line L2 is: (y2-0) (x+1)=(3-0) (0+1), y2=-3x-3

This picture is so messy, just look at the upstairs.

Asking such a question, how did you come to junior high school?

y=vertex coordinates (, axis of symmetry x=

Axis of symmetry x = 1

Intersection coordinates (1,2).

I can't read a word, we're in the sixth grade, what kind of mother's grade.

a(a,a)，b(b,8b)，a=ka+m

8b=kb+m

8b-a=k(b-a)

k=(8b-a) (b-a)=(8b a-1) (b a-1) so that b a=n, n is a positive integer.

k = (8n-1) (n-1) = (8n-8+7) (n-1) = 8+7 (n-1), obviously n is not equal to 1

So n is equal to 2 or 8, k = 15 when n = 2, and k = 9 when n = 8