What is a tick function and what are the properties

Tick function. nike function) is similar to an inverse proportional function.

is a function of the form f(x)=ax+b x(a>0,b>0). Named after the image, it is also known as the "double-hook function."

Tick function","Checkmark function.

Double delphinium function" and so on. Because the image of the function is similar to the Nike trademark, it is also called the "Nike function."

or the "Nike Curve".

Quality. Image.

The image of the tick function is two curves with y-axis and y=ax as the asymptote lines, and the product of the distance from any point to the two asymptotic lines on the image is exactly the sine value and | of the angle between the asymptote linesb|The product of .

If a>0 and b>0, in the first quadrant, its turning point is

Best value. When the defined field is , (a>0, b>0) takes the minimum value at , and the minimum value is

When the domain is defined as , the function has no maximum value, and when the domain is defined as , (a>0,b>0) takes the maximum value at , and the maximum value is .

Parity monotonicity.

Parity. The checkmark function is an odd function.

Monotonnia. Let k= , then:

Interval increase: and; Minus Interval: And.

Trend: first increase and then decrease on the left side of the y-axis, first decrease and then increase on the right side of the y-axis.

The tick function is a general function similar to the inverse proportional function, also known as the "double hook function" and "tick function"."Checkmark function. "Double delphinium function" and so on. It is also known as the "Nike Function" or "Nike Curve".

The so-called tick function (hyperbolic function) is a function of the form f(x)=ax+b x(a>0). Named from the image.

Image. Image properties of the tick function:

The tick function is a common and special function in mathematics, as shown in the diagram, it is best to draw the asymptote line y=ax when plotting.

Parity monotonicity.

When x > 0, f(x)=ax+b x has a minimum value (here, for the sake of research convenience, a>0, b>0 are specified), that is, when x=sqrt(b a) (sqrt means finding the quadratic root).

The checkmark function is a general hyperbolic function similar to the inverse proportional function, which is a function of the form f(x)=ax+b x(a>0, b is greater than 0).

In fact, the general form of the checkmark function is:

f(x)=ax+b x(a>0) However, in advanced mathematics, a is mostly only 1, and the value Chinese of b is indefinite. Science math changes are more complex.

Define the domain as (- 0) (0,+.)

The range is (- 2 ab] [2 ab,+

When x>0, there is x=root, b, and a, and the minimum value is 2<0, and x=- is b, and the maximum value is: The analytic expression of the 2-root-ab checkmark function is y=x+a x (where a>0), and its monotonicity is discussed as follows:

Set x1< x2, then f(x1)-f(x2)=x1+a x1-(x2+a x2)=(x1-x2)+a(x2-x1) (x1x2)=[(x1-x2)(x1x2-a)] (x1x2)< p>

The properties of the tick function are as follows:

1. The image of the checkmark function is two curves with y-axis and y=ax as the asymptote respectively, and the product of the distance from any point to the two asymptotic lines on the image is exactly the sine value and | of the angle between the asymptote (0-180°).b|The product of .

2. The tick function is an odd function.

3. Increase interval: and; Subtract interval: and {x|04. Change trend: increase first and then decrease on the left side of the y-axis, and decrease first and then increase on the right side of the y-axis.

Introduction to the Tick Function:

The image of the checkmark function is two curves with y-axis and y=ax as the asymptote lines, and the product of the distance from any point to the two asymptotic lines on the image is exactly the sine value and | of the angle between the asymptote lines (0-180°).b|The product of .

If a>0 and b>0, in the first quadrant, its turning point is [(b a) (1 2),2(ab) (1 2)]. The first derivative of the checkmark function: y'=-b/x^2+a。Parity: Odd functions.

The knowledge points of the tick function are summarized as follows:

1. The checkmark function is also known as the "checkmark function", "double hook function" and "checkmark function".

Expression: y=x+p x

When the function expression is y=qx+p x, we can extract q and make it y=q(x+p qx), so that the function can still be observed by nature.

2. Functional properties:

1) Parity.

When p>0, its image is distributed.

The two parabolas in the first and third quadrants cannot intersect with the x-axis and y-axis, and hail is an odd function.

When p>0, its image is distributed.

2. The two parabolas of the fourth quadrant cannot intersect with the x-axis and y-axis, which are also odd functions.

2) Monotonicity.

For the case of the first quadrant: with ( p,2 p) as the vertex, on (0, p] is the decreasing function, on [ p,+ is the increasing function, the opening is up;

In the third quadrant, (-p, -2 p) is the vertex, at (p], is the increasing function, and at [- p, 0) is the decreasing function, and the opening is downward. where the ordinates of the vertices are obtained by using the mean inequality for the function.

3. It is worth noting that in the image of the first quadrant, when x is smaller, that is, closer to 0, the left side of the image tends to the y-axis +, but does not intersect; When x is larger, i.e., tends to +, the right side of the image is closer to the straight line y=x half-branch, but does not intersect.

4. In the same way, in the third quadrant of the image, when the x is larger, that is, the closer to 0, the right side of the image tends to the y-axis, but does not intersect; When x is smaller, i.e., tends to - the closer the left side of the image is to the straight line y=x negative half-branch, but not intersecting. That is, the asymptote has a y-axis, and the straight line y=x.

5. Maximum: The method of finding the maximum: one is to use the monotonicity of the function, the second is the mean inequality, and the third is the special monotonicity, such as finding the maximum value of the function y=(x+5) x+4).

The property of the tick function: The image of the tick function is two curves with y-axis and y=ax as the asymptote lines, respectively, and the product of the distance from any point to the two asymptotic lines on the image is exactly the sine value of the angle of the asymptote (0-180°) and b|The product of . When the domain is defined, the function has no maximum; The checkmark function is an odd function.

The checkmark function is a general hyperbolic function similar to the inverse proportional function, which is a function of the form f(x)=ax+b x(ab>0). Named after the image, it is also known as "double hook function", "hook function", "check mark function", "double delphinium function", etc. Common a=b=1.

Because the image of the function is similar to the Nike logo, it is also referred to as the "Nike Function" or "Nike Curve".

The image of the tick function is two curves with y-axis and y=ax as the asymptote lines, and the product of the distance from any point to the two asymptotic lines on the image is exactly the sine value of the asymptote angle (0-180°) and b|The product of . If a>0 and b>0, in the first quadrant, its turning point is [(b a) (1 2),2(ab) (1 2)]. The first derivative of the tick function:

y'=-b/x^2+a。Parity: Odd functions.

The checkmark function is a general hyperbolic function similar to the inverse proportional function, which is a function of the form f(x)=ax+b x(ab>0). Named after the image, it is also known as the "double hook function", "hook function", etc"Checkmark function. "Double delphinium function" and so on. Common a=b=1.

The image of the checkmark function is two curves with y-axis and y=ax as the asymptote lines, and the product of the distance from any point to the two asymptotic lines on the image is exactly the sine value and | of the angle between the asymptote lines (0-180°).b|The product of .

A function of the form f(x)=ax+b x(a>0,b>0) is called a checkmark function.

The checkmark function is a general hyperbolic function similar to the inverse proportional function, which is a function of the form f(x)=ax+b x(ab>0). Named after the image, it is also known as the "double hook function", "hook function", etc"Checkmark function. "Double delphinium function" and so on. Common a=b=1.

Because the image of the function is similar to the Nike logo, it is also referred to as the "Nike Function" or "Nike Curve".

The image of the checkmark function is two curves with y-axis and y=ax as the asymptote lines, and the product of the distance from any point to the two asymptotic lines on the image is exactly the sine value and | of the angle between the asymptote lines (0-180°).b|The product of . If a>0 and b>0, in the first quadrant, its turning point is [(b a) (1 2),2(ab) (1 2)]. The first derivative of the tick function:

y'=-b/x^2+a。