How is the left addition and right subtraction in the function translation obtained?

You can understand it this way, after translating a unit to the left, the abscissa x1 obtained by adding a to the coordinates of this point is suitable for the original function, because this point is obtained by translating a unit to the left of the original function, so its abscissa is smaller than the point on the original function, so x1=x-a

That is, x=x1+a, and the point x is on the original function image, and the function about x1 can be obtained by substituting it, that is, the analytic formula of the new function obtained after translation. Because the argument of the function does not affect the letter (x, x1) no matter what letter is taken, it is enough to replace x1 with x.

In the same way, it can be understood as panning to the right, but it becomes x=x1-a

The translation of a function is to add a number or a number to x, and you have learned about vectors.

Let's set the translation a unit (a is a positive number), if you translate to the left, the translation vector p1=(-a,0); To the right, p2=(a,0).

Now the original point a(x,y).

Pan by vector p.

a‘=a+p

To the left: a'=a+p1=(x-a,y).

Due to a'(x',y')

There is x=a+x'

y=y'The original function y=f(x).

After translation: y'=f(a+x') is equivalent to y=f(x+a) This is not added to the left...

Right minus the same.

Take the primary function y=2x as an example, 1When the function is shifted upwards by one unit, x does not change, and y becomes larger by 1, at which point y=2x+1.

2.When the function is shifted down by one unit, x does not change, and y becomes smaller by 1, so y=2x-1.

3.When the function is shifted one unit to the left, y does not change, and x becomes smaller by 1, in order to keep the value of y constant, x needs to increase by 1, and y = 2 (x+1).

4.When the function is shifted one unit to the right, y does not change, x becomes larger by 1, and in order to keep the value of y constant, x decreases by 1, and y = 2 (x-1).

Why is the derivation step different from what it seems?

Actually, it's the same:

When the function is shifted one unit upwards, x does not change, and y becomes larger by 1, in order to keep x unchanged, y should be reduced by 1, at this time y-1=2x, and the constant term 1 is moved to the right, at this time y=2x+1.

When the function is shifted down by one unit, x is unchanged, y becomes smaller by 1, in order to keep x unchanged, y should be increased by 1, at this time y+1=2x, and the constant term 1 is moved to the right, at this time y=2x-1.

You can think of it this way: the left and right shifts are for x, if you move to the right, the value of x' after the move will be larger than the value of the original x, and x'-a = x (a is the distance to the right), and after the move, find the function of y about x', that is, the function of y about (x'-a). The same is true for moving to the left, except that a is a negative number, i.e. x' is smaller than x, a, x'+a=x.

This is the mantra for the properties of function translation, and the specific principle is as follows.

1. When the function image is translated to the left and right, the ordinate remains unchanged, and the abscissa follows the rule of left addition and right subtraction; When the function image is translated upwards and downwards, the abscissa remains the same, and the ordinate follows the rule of subtracting from the top.

2. The function image translation is nothing more than two situations, that is, the left and right translation and the upper and lower flat Ming manuscript shift. The left and right translations of the function image are for the abscissa x, and the up and down translations of the function image are for the ordinate y.

The essence of the translation of the function image is the movement of the position of the function image, and the function image itself does not change, but the corresponding coordinates of the translated function image in the two-dimensional coordinate system have changed.

Function Translation Method:

The display function y=f(x) adds left and subtracts right, and adds up and subtracts.

The function f(x) is shifted to the left by a unit, and the resulting function is g(x)=f(x+a). To the right is g(x)=f(x-a).

The function f(x) translates a unit upwards and the resulting function is g(x)=f(x)+a. Downwards is g(x)=f(x)-a.

For example, the function is y=a(x-h) +k, the left addition and right subtraction are the addition and subtraction on h, and the addition and subtraction are the addition and subtraction on the absence and k.

For the x and y terms of the implicit function, the positive direction is subtracted (the positive direction of the coordinate axis).

For example, the quadratic function y=ax +bx+c translates a unit to the right and then upwards to get (y-b)=a(x-a) +b(x-a)+c and then finishes.

For example, the ellipse x a +y b =1 translates a unit to the left and then translates b units downwards to get (x+a) a +(y+b) b =1 and then finishes.

Summary. Hello, I'm glad to answer for you You can understand that the abscissa of x1 is obtained after translating a unit to the left, and the abscissa x obtained by adding a to the coordinates of this point is suitable for the original function, because this point is obtained after the original function translates a unit to the left, so his abscissa is smaller than the point a on the original function, so x1=x-a is x=x1+a, and the point x is on the original function image, and the function about x1 can be obtained by substituting it. That is, the analytic formula of the new function obtained after translation. Because the independent variable of the function does not affect no matter what letter (x, x1) is taken, it is enough to replace x1 with x, and the same can be understood as translating to the right, but it becomes x=x1-a

Why is the function image panning in math "left plus right minus" instead of "left minus right plus"?

Hello, I'm glad to answer for you You can understand it this way, the point with the abscissa x1 obtained after translating a unit to the left, and the abscissa x obtained by adding a to the coordinates of this point is suitable for the original function, because this point is obtained after the original function translates a unit to the left, so his abscissa is smaller than the point on the original function, so x1=x-a is x=x1+a, and the point x is on the original function image, and the function about x1 can be obtained by substituting it. That is, the analytic formula of the new function obtained after translation. Because the independent variable of the function does not affect (x, x1) no matter what word is taken, it can be replaced with x1 by x, and the same can be understood as translating to the right, but it becomes x=x1-a

For example, hail y=x 2 passes through the origin of the annihilation faction (0,0).The shift to the left is y=(x+1) 2, because the intersection point of the function with the x-axis is (-1,0) after the leftward translation, so when x=-1, the value of the function is 0, so the formula becomes y=(x+1) 2 In the same way, if the root of the function becomes positive after it is turned to the right, that is, (x-1) 2=0 has a solution only when x=1, and the reaction to the function formula becomes left addition and right subtraction.

I describe this idea in words, quadratic functions.

In fact, it describes the relationship between y and x.

Look at y=x in the most simplified way

When y=0, x=0

The relationship between the two of them is very straightforward.

Now look at the translation, the translation is actually to let you find the relationship again, for example, to the left to shift 2, which means that when y remains unchanged at 0, the x value corresponds to -2. At this time, the left and right sides of the relationship become 0=-2. The relationship between x and y is definitely wrong, so we need to re-find the relationship between the two, that is, to rebalance both sides of the equation.

In this case, y=x+2 is the relation of xy (0=-2+2). It is the parabola after the quadratic function is translated.

The same goes for panning to the right. y remains the same, the corresponding x has an extra 2, and 2 must be subtracted to the right side of the equation to keep the flat and vertical equation

Let's put it in layman's terms: an apple on each side of the scale, and the balance is balanced. Someone has changed the apple on the right to a banana (shifted to the left), and he wants to balance the scales when he is done, then he must add something more to the banana.

To add a little more, y=ax 2 we can think of it as the essence of the gods. y=a(x-h) 2 and y=ax 2+k are the left and right phantom clones and the upper and lower phantom clones of the body. It's intuitive.

Why x is left plus right and y is up and down minus.

Because the left and right shifts correspond to x, the value of the comic x changes and the result y does not change, which leads to the necessity of changing the relation. Moving up and down is just a constant and y, and x is fixed. Constant plus, y plus. Constant subtraction, y subtraction.

The vertex coordinates of the quadratic function y x are at the origin. Let's look at the quadratic function y (x-m) and apply the minimum value to find the vertex coordinates. Let dy dx 0 to get the vertex coordinates (m,0), and obviously the head of the figure is shifted to the right m.

For example, y (x m), the vertex coordinates are (-m,0), and the vertices of the image are shifted to the left by m. As the so-called (every left shift, the first number of children every - right shift). The same is true for impulse functions, e.g. δ(x-2), where the pulse occurs at the position of x2.

Now the function y (x-h) k, the vertex coordinates of the image (h, k) are discussed. Looking at the function y (x-h-m) k, find the extremum: let dy dx 0,2(x-h-m) 0,x h m, the vertex coordinates are (h m, Qi collapse k), and compare the coordinates of the two vertices to see that the vertex has shifted to the right.

The left addition and right subtraction of the quadratic function are for x, and the addition and subtraction of the quadratic function are also for x, not others.

This is a question of what to refer to.

Treat it as a graph translation: move left to add, right to subtract.

If it is regarded as a coordinate translation, the left shift of the coordinates is subtracted, and the left Yu Hu is moved to the right is added.

Well, I hate rote memorization too.

For y=x, move one unit to the left.

Get y=x+1. Why?

Because of the shift to the left, the intersection point on the x-axis also changes, and it must be x+1, and y is equal to 0, that is, y is essentially unchanged, but x has changed, so it is left plus right subtraction.

If you shift the parabola one unit to the left, then the abscissa of all points will be -1. The ordinate remains unchanged and the abscissa decreases. The same y, originally equal to x (y x).

Now, to make x smaller, add a number to the parentheses, which becomes y (x n).

Conversely, to make x bigger, subtract a number.

Let's take the straight line y=x as an example.

Straight line (when the line is translated one unit to the left, then (-1,0) means that y=(x-(-1)).

This would make it easy to understand why y=x+1

When the line is translated 1 unit to the right, it is translated from the origin to the positive half axis of x. Is the straight line going to be over the point (

In this case, the equation is y=x-1

When the straight line is translated upwards by 1 unit, the straight line is shifted upwards by one unit from the y-axis, and the straight line passes (

y-1=x is y=x+1

If it moves down, it is y+1=x, so y=x-1

It's really not good to draw a diagram for this kind of problem, and the same is true for quadratic functions.

Seek to adopt [Star Eyes].

I can't ask again

Ask for the adoption of the [Persistent Face].

For example, y=x+2( ) wants to move two units to the left, but the translation y to the left must not change, right? According to the normal way of thinking, x should be minus 2, right? But then you will find that y=x+2 becomes y=x( ) At this time, let's say you want to move the point of x=0 and let it translate 2 squares to the left, and the result should be -2, but you will find that the value of y corresponding to -2 in the analytic equation ( ) that you get by subtracting 2 is -4, so at this time, -4≠0, y≠ in y, so it is not true, then in order to become y y and x -2, you need to translate 4 units upwards to make y y, so y x+2-2+4, y x+4 is also equal to (x in the above analytical) plus 2.

Mathematics for junior 2: Why is the translation of a function added left and subtracted from the right?

For example, if f(x+5) and f(x) are the same, the x in f(x+5) is less than 5 compared with x in f(x), because (x+5) is equivalent to (x) as a whole, from the perspective of defining the domain, let f(x) define the domain d, then f(x+5) satisfies x+5 on d, then the definition domain of f(x+5) (which is the range of the independent variable x, and the formula in parentheses is on d) is subtracted by 5 on the basis of d, i.e., f(x+5) The definition domain is f(x) The definition field is shifted to the left by 5 unit lengths, and the function image is also the same as the translation direction and translation length of the definition domain, otherwise the definition domain is not satisfied. (For example: f(x)=x, then f(x+5)=x+5, if you look at the image f(x+5), f(x) is f(x) to the left to translate 5 units of length.) ）

Let's take a look at a simple function.

The principle of function translation, left addition and right subtraction refers to the left-right movement of the function image in the plane coordinate system by changing the independent variables of the function. Specifically, when an independent variable is added with a positive number, the function image is shifted to the left; And when the argument subtracts a positive number, the function image shifts to the right. The following describes the principle of function translation, left addition and right subtraction according to the serial number heading.

1.Overview of function panning:

Function translation refers to the operation of moving the function image left and right along the horizontal axis (x-axis) in the coordinate system. Translation changes the position of a function so that it moves horizontally relative to its original position. The principle of function translation, left addition and right subtraction, refers to the left and right translation of the function image by adding or subtracting the independent variables.

2.Left-plus translation principle:

When a positive number is added to the independent variable of a function, the function image is shifted to the left. This is because in a function, an increase in the independent variable leads to a decrease in the value of the function. Therefore, when a positive number is added to the independent variable, the function image shifts to the left in the coordinate system, i.e., the overall shift is in the negative direction.

3.Right Subtraction Translation Principle:

Conversely, when a positive number is subtracted from the function's independent variable, the function image shifts to the right. This is because in a function, a decrease in the argument results in an increase in the value of the function. Thus, when an independent variable subtracts a positive number, the function image shifts to the right in the coordinate system, i.e., the whole is offset in the positive direction.

4.Effect of panning:

Function translation has a certain effect on the shape and characteristics of the function. Translation does not change the slope and curvature of the function, but it does change the position of the function and the horizontal position of the image in the coordinate system. By adding left and subtracting the arguments, you can move the function image anywhere to make it more suitable.

5.Representation of translation:

Function translation can be expressed in terms of the definition of a function. For example, for the general function f(x), if you want to shift the function image to the left h units, you can define the function as f(x-h).

Similarly, if you want to shift the function image to the right by h units, you can define the function as f(x+h). In this way, by changing the definition of the function, the function image can be shifted left and right.

Summary: The principle of function translation left addition and right subtraction is to realize the left and right movement of the function image in the coordinate system by adding or subtracting the independent variables of the function.

Adding a left pan shifts the function image to the left, and subtracting the right panning shifts the function image to the right. Translation can change the position of the function, but not its slope and curvature. By changing the definition of a function, you can flexibly translate the function.