Function symmetry Summary

 

This article focuses on the monotonicity, parity, and periodicity of functions. However, there are many questions about function symmetry, continuity, and concave and convex. In particular, symmetry, because the textbook has a scattered introduction to it, such as the symmetry axis of the quadratic function, the symmetry of the inverse proportional function, and the symmetry of the trigonometric function, the frequency of examination has been relatively high. From the author's experience, this aspect has always been a difficult point in teaching, especially the symmetry judgment of abstract functions. Therefore, let's make a rough summary of the involved functional symmetry knowledge.

I. Concept of symmetry and symmetry of common functions

1. Concept of symmetry ① Function symmetry: if the image of a function is folded along a straight line, the image on both sides of the line can completely overlap, it is said that the function has symmetric symmetry, this line is called the symmetry axis of the function. ② Center symmetry: if the image of a function rotates 180 degrees along a point and the image obtained completely overlaps with the original function image, the function is called Symmetric center symmetry, this point is called the symmetric center of the function.

2. Symmetry of common functions (all function independent variables can have all meaningful values)

① Constant function: it is symmetric and center symmetric. All vertices on a straight line are symmetric centers of the line, and the straight line perpendicular to the line is its symmetric axis.

② Symmetric function: it is symmetric and center symmetric. All vertices on a straight line are symmetric centers of the line, and the straight line perpendicular to the line is its symmetric axis.

③ Quadratic function: it is axial symmetry, not center symmetry. Its symmetry axis equation is X =-B/(2a ).

④ Inverse proportional function: it is symmetric and center symmetric, where the origin is its symmetric center, and y = x and y =-X are its symmetric axes.

⑤ Exponential function: it is neither axial symmetry nor central symmetry.

⑥ Logarithm function: neither axial symmetry nor central symmetry.

7. Power function: Obviously, the odd functions in the power function are symmetric in the center, the symmetric center is the origin, the even functions in the power function are symmetric, And the symmetric axis is the Y axis. Other power functions do not have symmetry.

Cosine sine function: it is symmetric and symmetric. (K π, 0) is its symmetric center, and X = K π + π/2 is its symmetric axis.

Cosine sine function: the sine function y = Asin (ω x + PHI) is both an axial symmetry and a central symmetry. You only need to extract X from ω x + Phi = K π, that is, the abscissa of its symmetric center. The ordinate is of course zero. The X from ω x + Phi = K π + π/2 is its symmetric axis; it should be noted that if the image is moved up to the top of each grade of High School Courseware Teaching Case exercises, and the physical chemistry of Chinese mathematics and English is moved down, the axis of symmetry will not change, but the ordinate coordinates of the symmetric center will change.

Cosine cosine function: it is symmetric and center symmetric. x = K π is its symmetric axis, and (k π + π/2, 0) is its symmetric center.

 

11. tangent function: not symmetric, but symmetric in the center, where (K π/2, 0) is its symmetric center, which is prone to mistakes is that some colleagues may mistakenly think that the symmetric center is only (K π, 0 ).

12. logarithm function: the logarithm function y = x + A/X (where a> 0) is symmetric in the center because it is a strange function, and the origin is its symmetric center. But what is easy to make mistakes is that students may mistakenly think that the most important value is its symmetric axis. For example, when processing the Function Y = x + 1/X, they mistakenly think that there will be f0.5) = f (1.5 ), when I was teaching, I always asked students: Have you ever seen a teacher draw the "√" on both sides? The students immediately understand and remember deeply.

13 cubic functions: Obviously, the odd functions in cubic functions are symmetric in the center, and the symmetric center is the origin. However, the symmetry of other cubic functions varies depending on the question.

14 absolute value function: here we mainly talk about y = f (│ x │) and Y = │ f (x) │. The former is obviously an even function, which will be symmetric on the Y axis; the latter is to symmetric the image below the X axis to the top of the X axis, whether it still has symmetry, there is no certain conclusion, for example, y = │ lnx │ has no symmetry, while y = │ SiNx │ is still symmetrical.

Ii. Symmetry prediction of functions

1. Special symmetry prediction for specific functions

① A function is generally not related to the X axis, which is determined by the function definition, because an X does not correspond to two y values. However, we can extend a curve here, which may be symmetric on the X axis. Example 1 judge the symmetry of the curve y ^ 2 = 4x.

② About the y-axis symmetry of the function Example 2 judge the symmetry of the Function Y = cos (sin (x.

③ Example 3 of the symmetry of the function about the Origin judgment Function Y = (x ^ 3) × SiNx symmetry.

④ Function about y = x symmetry Example 4 judge the symmetry of Function Y = 1/X.

⑤ Function about y =-x symmetry Example 5 judge the symmetry of the Function Y =-4/X.

In my summary, set (x, y) to any point in the original curve image. If (x,-Y) is also on the image, the curve is symmetric on the X axis; if (-X, Y) is also on the image, the curve is about y axis symmetry; If (-X,-Y) is also on the image, the curve is about the Origin symmetry; if (Y, x) is also on the image, the curve is symmetric with Y = x. If (-y,-x) is also on the image, the Y =-X axis is symmetric.

2. Symmetry prediction of Abstract Functions

① Axial symmetry

Example 6 if the Function Y = f (x) satisfies f (x + 1) = f (4-x), obtain all the symmetry axes of the function.

(If any value is substituted into x = 0, F (1) = f (4) and the center is 2.5, the function is about x = 2.5 symmetric)

Example 7 if the Function Y = f (x) satisfies f (x) = f (-x), obtain all the symmetry axes of the function.

(In the same way as in the above example, we can guess that the axis of symmetry is x = 0. It can be seen that the even function is a special axial symmetry)

Example 8 if f (x) is an even function and f (x + 1) = f (x + 3), find all the symmetry axes of the function.

(Because f (x + 1) = f (-X-3), according to the above example, we can guess the axis X =-1, because it takes 2 as the cycle, so X = K is all its axes of symmetry, K, Z)

② Center Symmetry

Example 9 if the Function Y = f (x) satisfies f (3 + x) + f (4-x) = 6, obtain the symmetric center of the function.

(Because when the sum of the independent variables is 7, the sum of the function values is always 6, so the midpoint is fixed to (3.5, 3), which is its symmetric Center)

Example 10 if the Function Y = f (x) satisfies f (-x) + f (x) = 0, calculate all the symmetric centers of the function.

(We can guess that the symmetric center is (0, 0) by using the same method as in the preceding example. It can be seen that the odd function is a special center symmetry)

Example 11 if f (x) is a strange function and f (x + 1) + f (x + 3) is 0, all symmetric centers and axes of the function are obtained.

(The periodic frequency is defined as 4, and f (x + 1) =-f (x + 3), so f (x + 1) = f (-X-3 ), as shown in the preceding example, x =-1 is the symmetric axis, so x =-1 + 2n is the symmetric axis, (2 k, 0) is the symmetric center, where K, Z)

My summary is:

① When the symbols in front of X in parentheses are positive and negative, they tell us the symmetry. We can use special values to guess the number of pairs called. Here we do not advocate conclusion, because it is easy to confuse with the following conclusions.

② When the symbols in front of X tell us at the same time that they are cyclical. For example f (x + 1) = f (X-5) is telling us it takes 6 as the cycle.

③ When the symbols in front of X are the same and tell us parity, we can also introduce symmetry because parity has the ability to create negative signs. 3. the symmetry equations of Y = f (x + 2) and Y = f () are obtained in the symmetry prediction example of two abstract functions. (When X of the first function is 0, the value is F (2). At this time, the X of the second function must be-1 to correspond to so many, their center is-1.5, So we guess the axis of symmetry is X =-1.5)

My summary is:

① When the symbols in front of X in parentheses are positive and negative, they tell us the symmetry. We can still guess the number of pairs called by using special values. Here we still do not advocate conclusion, because it is easy to confuse with the previous conclusion.

② When the symbols before X tell us at the same time that it is image translation. For example, y = f (x + 2) and Y = f (x-1), the former is obtained by moving the latter three units to the left.

Iii. Proof of symmetry: it is not enough to guess the conclusion when answering a major question. We need to provide a complete proof.

1. Symmetry proof of a function

Example 13 proves that if the Function Y = f (x) satisfies f (a + x) = f (B-x), The Function) /2 symmetric.

Proof: If y = f (x) is used to take the vertex (m, n), then n = f (M), and the vertex (m, n) the symmetry of X = (a + B)/2 is (a + B-m, n), and f (a + B-m) = f (a + (B-m) = f (B-m) = f (m) = N, which indicates (A + B-m, n) is also in the original function image, so that the original function is symmetric With Line X = (a + B)/2.

In my summary, the core is the indirect method, that is, taking a point in the function. If the symmetry point is still on the function image, we can conclude that the function is symmetric about it.

2. Proof of symmetry between two functions

Example 14 demonstrates that the Y = f (a + x) function is symmetric with the Y = f (B-x) function about linear x = (B-a)/2. (Note that it is not (a-B)/2. The method proved is similar to the method in the previous example)

In my summary, it is still an indirect method, but once more, you need to take a point in the function. If the symmetric point is on the image of the other function, take a point in the other function at the same time, the symmetry point is in the function image, so we can conclude that the function is symmetric. The reason for taking the image twice is to prevent the two images from being a part of the symmetric image.

3. In particular, the proof of Y = x symmetry example 15 proves that Y = (2x + 1)/(3x-2) is about y = x symmetry.

(You only need to find its own inverse function)

In my summary, ① A function itself does not need to use the indirect method of Y = x symmetry. It only needs to prove that its inverse function is its own. ② The two functions do not need to use the above tedious method to prove the Y = x symmetry. They only need to prove that the two functions are inverse functions, that is, finding one inverse function as another. ③ This sentence is also true. To prove that the two functions are inverse functions, you only need to prove that their image is symmetric with Y = x.

Iv. Application of symmetry

1. Evaluate Example 16 F (x) = 4 ^ x/(4 ^ x + 1) known, evaluate F (-4) + f (-3) + f (-2) + f (-1) + f (0) + F (1) + F (2) + f (3) + f (4. (We only need to consider whether the sum of the function values is a fixed value when the two independent variables add up to 0. What is clearly implied here is the symmetry of the function)

In my summary, the symmetry is mainly used to examine the relationship between a pair of function values.

2. "Symmetry + symmetry" can deduce Periodicity

Example 17 if the Function Y = f (x) satisfies the requirements of f (x + 3) = f (2-x) and F (4 + x) = f (5-x ), returns the Minimum Positive Period of the function. (Because f (x + 3) = f (2-x) = f (4 + (-2-x) = f (5-(-2-x) = f (7 + x) so the cycle is 4) I sum up: the two symmetry can be combined to symbolically form the same number, so as to obtain the periodicity.

3. Parity + symmetry can be used to deduce periodicity. As mentioned above, it is still because parity has the ability to create negative signs.

4. parity of trigonometric Functions

Example 18 if the Function Y = 3sin (2x + θ + π/4) (where 0 <θ <π) is a odd function, evaluate the value of θ.

(2x + θ + π/4 = K π, and x = 0, so θ + π/4 = K π, only θ = 3 π/4 in the required range)

In my summary, the parity of almost all trigonometric functions is used as symmetry. First, find all the symmetry axes, and then the Y axis is one of them (or first find all the symmetric centers, and the origin is one of them ).

5. Y = x symmetric applications

In Example 19, obtain the symmetric axis equations of the Functions f (x) = E ^ (x + 1) and g (x) = ln (x + 1.

(Because f (x) = E ^ X and g (x) = lnx are reciprocal functions, y = x is symmetric, and f (x) = E ^ (x + 1) is obtained by moving f (x) = E ^ X to the left, g (x) = ln (x + 1) is also obtained by G (x) = lnx is obtained by moving a unit to the left, so the axis of symmetry also follows a unit to the left, that is, y = x + 1)

6. The meaning of symmetry

Example 20 if y = as133 + bcosx is symmetrical about x = π/4, calculate the slope of the straight line AX + by + 3 = 0. (Since x = π/4 is symmetric, F (0) = f (π/2)

Function symmetry Summary