What s the deal with the checkmark function? To be specific in nature and application, it is necessa

The checkmark function is a function of the form y=ax+b x (a, b are not equal to 0), and has the following characteristics:

1.The checkmark function is obtained by hyperbolic rotation, so there are also asymptotes, focal points, vertices, and so on.

2.The checkmark function is always odd and is centrally symmetrical with respect to the origin.

3.The two asymptotic lines of the checkmark function are always y-axis and y=ax

4.When a, b, > 0, the images are distributed in the first place.

The part between the two acute angles of the two asymptotic lines of the first quadrant of the first quadrant is discussed only because of its symmetry. Using the important inequality, we can see that the minimum value is 2 root number ab, which is obtained when x = b a under the root number, so it decreases monotonically on (0, b a under the root number) and increases monotonically on (b a under the root number, positive infinity).

5.When a>0 and b<0, the image is distributed between the obtuse angles of the four quadrants and the two asymptotic lines, and the two branches are monotonically increasing and have no extremes.

Other cases can be obtained by transforming.

In short, it is very easy to make an image of the checkmark function, remember that it is hyperbola, then make an asymptote, and then find a special point, and you can make the whole image.

As for how to judge the monotonicity of the sign function, it can be proved by the definition method, or the positive and negative signs can be judged by the derivative, the latter is simpler.

For example: y=ax+b x, a>0, b>0;

The derivative proves that at x>0, it first decreases monotonically, then increases monotonically, and at the lowest point in the middle, x = the root number (b a).It's like a check mark.

At x<0, from 0 to minus infinity, is an inverted checkmark, i.e., it increases monotonically and then decreases monotonically, and the maximum value is x = negative root (b a).

1.The checkmark function is obtained by hyperbolic rotation, so the noisy brigade also has asymptotto, focus, vertices, and so on.

2.The number of the town is always an odd function, and the center of the stool is symmetrical with respect to the origin.

3.The two progressive lines of the checkmark function are always y-axis and y=ax, and I know so much.

"check" function, expression f(x)=x+a x (a>0);

Nature (1) Define the domain.

2) f(x) is an odd function, i.e., the image is symmetrical with respect to the origin, (3) because f(x)=1-a x, so f(x) is a subtraction function at x (0, a), and an increasing function at x (a, + positive infinity);

f(x) is an increasing function at x (negative infinity, -a) and a decreasing function at x(-a, 0).

4) At x 0, when x = a = 2

At x 0, when x = - a = -2

Is there such a function as a checkmark function?

A function of the form y=ax+b x+c (ab>0, c is a constant) can be called a checkmark function. It is named because its image resembles two check marks. Here I will introduce some properties of the function when a>0, b>0, c=0.

Domains: Increasing intervals: (-infinity, -sqr(b a)) and (sqr(b a), + infinity).

Subtraction intervals: (-sqr(b a),0) and (0,sqr(b a)).

Extreme point: sqr(b a).

Parity: Odd functions.

It should be noted that on one branch of the check function image, this one is not symmetrical with respect to x=sqr(b a), so be careful when using it.

The image at a<0,b<0 is symmetrical with respect to the y-axis at a>0,b>0; The image at c≠0 is obtained by translating the image at c=0 up and down. The nature of the two should be easily analogous, so I won't go into it here.