High School Math Logarithmic Functions Translation Problems

The left shift is the change of x, that is, the value of the same function x becomes smaller, the upper shift is the same abscissa, and the value of y is increased, the shape of the function is unchanged, and the position changes.

For example, moving an object from place A to place B does not change its shape, for example, the function y=lgx (1) passes the point (1,0).

Translate 2 units to the right to get y=lg(x+2) (2) This function passes the point (-1,0).

Translate (1) down by 1 unit to get y=(lgx)-2 (3) This function passes the point (1,-1).

Translate (2) upwards by 2 units to get y=(lg(x+2))+2 (4) This function passes the point (-1,2).

The shape of the same four curves, due to the coordinate system.

They are not the same function.

The translation satisfies the law, the left adds and the right subtracts, and the upper and lower subtracts.

For example, if the function y=lgx is shifted two units to the right, we get y=lg(x-2), and y=lgx is constantly passing through (1,0) points, and after translation, it is constantly passing (3,0);

If y=lgx is translated to the left by two units, y=lg(x+2) is obtained, and y=lgx is constantly passed through (1,0) points, and after translation, it is constantly passed (-1,0);

If y=lgx is translated upwards by two units, y=lgx+2 is obtained, and y=lgx is always past (1,0) points, and after translation, it is constant past (1,2);

If y=lgx is translated downward by two units, y=lgx-2 is obtained, and y=lgx is constantly passed through (1,0) points, and after translation, it is constantly passed (1,-2);

log2 (x squared -x) -10 I figured out the range, but what about this logarithmic image panning problem? It should be one unit down.

Which point did (1,0) become?

Analysis: log(2,x 2-x)-1 logarithmic function definition domain: x 2-x>0==>x1

log(2,x 2-x)x 2-x-2-1

The set of solutions to the inequality is -1

This inequality is solved without image panning.

Do add and subtract right (for x, add up and subtract for constants, the variable coefficient abscissa reduction factor is the reciprocal, and the function coefficient ordinate expands the function coefficient multiple.) The sequential abscissa is first translated and then expanded, and the ordinate is expanded first and then translated.

y=asin(wx+q)+b

Compare with y=sinx.

First of all, the x-coefficient of the sin function variable should be 1, y=asin[w(x+q w)]+b

Left plus right minus, here plus , it means that q w units have been moved to the left. Add up and subtract the constant, the constant here is to add b, so it is to translate b units upward. Here the variable coefficient variable is x and the coefficient is w, so the abscissa is reduced by a factor of 1 w.

The coefficient of the function refers to the sin function, and its coefficient is a, that is, the ordinate is expanded by a fold.

For example, sinx is changed to sin(2x-pie 6).

It is sinx that first translates 12 units, and then doubles the abscissa.

I've summarized it for you, and I'll do a few questions according to my method.

The first equation is to use the reverse method, and push it backwards. Tightly integrated with the second equation:

y=sinx, generally translate to y=sin2x first, that is, the x-axis is reduced by 2 times and then translated to y=sin2 (x-pie 12) (note: 2 must be mentioned here). Remember:

Add up and subtract down, add left and subtract right. If y=sinx becomes y=2sinx, then the y-axis is doubled, note the difference between them.

This is the trigonometric function used.

The algorithm between the trigonometric functions is cyclical.

There is periodicity, look at your first example, multiply 2 by y=cos(x-pie 3), which is y=cost

Here t=x-pie 3

It's like. Moved x to send 3 units.

Left plus right minus. t is the x in the result

The first equation is to use the reverse method, and push it backwards. Tightly integrated with the second equation:

y=sinx, generally translate to y=sin2x first, that is, the x-axis is reduced by 2 times and then translated to y=sin2 (x-pie 12) (note: 2 must be mentioned here). Remember:

Add up and subtract down, add left and subtract right. If y=sinx becomes y=2sinx, then the y-axis is doubled, note the difference between them.

This is the trigonometric function used.

The algorithm between the trigonometric functions is cyclical.

There is periodicity, look at your first example, multiply 2 by y=cos(x-pie 3), which is y=cost

Here t=x-pie 3

It's like. Moved x to send 3 units.

Left plus right minus. t is the x in the result

The practical significance of a function translation: It only represents a change in its relative position in the coordinate system (or coordinate plane), but has no effect on the properties of the function itself and the practical meaning of its representation. For example:

y=kx+b, moving up or down means that the entire line is translated up or down by several units along the direction of the y-axis.

Quadratic functions add left and subtract right plus up and down subtract.

Let the function be y=a(x-h) 2+k which is the vertex formula, then the left addition and right subtraction is the addition and subtraction on h, which refers to x.

Addition and subtraction is the addition and subtraction on k, which refers to y.

Generalization: The function f(x) translates a unit to the left, and the resulting function g(x) = f(x+a).

The function f(x) translates a unit upwards, and the resulting function g(x) = f(x) + a In short: the function translation formula:

Left plus right minus. Add minus below.

Instructions:1Left and right are for x, and up and down are for y.

Function translation is generally divided into three types of problems:1The analytic formula of the function obtained by the translation is obtained from the analytic formula of the known function and the translation of its image. 2.

The analytic formula of the function and the analytic formula of the function obtained after the translation of the image are known, and the translation of the function image is judged. 3.The analytic formula after the known translation situation and the translation is found before the translation.

The left and right translation of the function, the left plus the right minus, the up and down translation, the up and down subtraction, and the flipping of the image.

f(x)=log(a)(x),g(x)=log(a)(x)+b,x belongs to (0, positive infinity).

The range of f and g is r

Since x cannot be 0, then f,g do not intersect.

Because a>1, the graph knows that the intersection of f,g on the y-axis is at negative infinity.

You can think of negative infinity as a "bottomless pit", no matter how much you add it, it's negative infinity.

Since f intersects the y axis at negative infinity, then g and y also intersect at negative infinity, and +b does not work at negative infinity.

The significance of f(x)+b is that each point on f(x) is translated upwards without an intersection with the x and y axes.

You can think: if y belongs to (negative infinity, a), then y+b belongs to (negative infinity, a+b), and +b has no effect on negative infinity.

g(x) can be seen as f(x) translating b units upwards.

Translations and intersections are not associated.

1.Take any point (x0,y0) on the y=f(x) image, and let its symmetry point about x=a be (x,y), then x+x0=2a, y=y0, that is, x0=2a-x, y0=y, substitute it into y0=f(x0), then y=f(2a-x).

2.When the abscissa of two points is equal and the ordinate is opposite, it is symmetrical with respect to the x-axis; When the ordinates are equal and the abscissa is opposite, the symmetry is with respect to the y-axis; All are symmetrical when they are symmetrical with respect to the origin.