MindMap Gallery function
A super detailed mind map of high school mathematics functions, including the concept and representation of functions, Monotonicity and maximum value, parity, symmetry and periodicity, graph of functions, etc.
Edited at 2024-02-05 16:20:35Avatar 3 centers on the Sully family, showcasing the internal rift caused by the sacrifice of their eldest son, and their alliance with other tribes on Pandora against the external conflict of the Ashbringers, who adhere to the philosophy of fire and are allied with humans. It explores the grand themes of family, faith, and survival.
This article discusses the Easter eggs and homages in Zootopia 2 that you may have discovered. The main content includes: character and archetype Easter eggs, cinematic universe crossover Easter eggs, animal ecology and behavior references, symbol and metaphor Easter eggs, social satire and brand allusions, and emotional storylines and sequel foreshadowing.
[Zootopia Character Relationship Chart] The idealistic rabbit police officer Judy and the cynical fox conman Nick form a charmingly contrasting duo, rising from street hustlers to become Zootopia police officers!
Avatar 3 centers on the Sully family, showcasing the internal rift caused by the sacrifice of their eldest son, and their alliance with other tribes on Pandora against the external conflict of the Ashbringers, who adhere to the philosophy of fire and are allied with humans. It explores the grand themes of family, faith, and survival.
This article discusses the Easter eggs and homages in Zootopia 2 that you may have discovered. The main content includes: character and archetype Easter eggs, cinematic universe crossover Easter eggs, animal ecology and behavior references, symbol and metaphor Easter eggs, social satire and brand allusions, and emotional storylines and sequel foreshadowing.
[Zootopia Character Relationship Chart] The idealistic rabbit police officer Judy and the cynical fox conman Nick form a charmingly contrasting duo, rising from street hustlers to become Zootopia police officers!
function
The concept and representation of functions
definition
three elements of function
Domain
Find conditions that limit the value of x
Non-empty number set A
Fractional function
The denominator is not 0
even square root function
Odd root radicand ∈ R
radicand ≥ 0
logarithmic
real number>0
range
The range is the set of y values corresponding to the x value
The value range must be a subset of set B, not necessarily set B itself.
separation constant method
inverse solution
Number-shape combination method
monotonicity method
Derivative method
Correspondence
function representation
analytical method
matching method
substitution method
undetermined coefficient method
Equation thinking
image method
List method
Monotonicity and Maximum
Monotonicity of a function
definition
increasing function
subtract function
How to judge monotonicity
Definition
Image method
Taking advantage of the monotonicity of known functions
Derivative method
The monotonicity of functions in two different intervals is the same. Generally, they are written separately and do not connect with "," or "and". Do not use ∪
Common conclusions about monotonicity
increasing function increasing function = increasing function
increasing function - decreasing function = increasing function
Decreasing function Decreasing function = Decreasing function
Decreasing function - increasing function = decreasing function
Monotonicity of composite functions
Same increase and different decrease
within the public domain
The maximum value of the function
definition
Basic method to find the maximum value of a function
functional method
monotonicity method
Image method
basic inequality method
Parity, symmetry and periodicity
Parity
definition
The domain is symmetric about the origin
odd function
even function
It is both an odd function and an even function
non-odd and non-even functions
The domain is not symmetric about the origin
Image features
odd function
The image is symmetrical about the origin
even function
The image is symmetrical about the y-axis
Common conclusions about parity
odd function odd function = odd function
even function even function = even function
Odd function × odd function = even function
Even function × even function = even function
Odd function × even function = odd function
cyclical
definition
minimum positive period
Common conclusions about periodicity
The constant difference between brackets is periodic
periodic function
symmetry
Symmetry of a function
It is a symmetry that the sum inside the brackets is a constant
Symmetry between two functions
Common conclusions about symmetry
Second-level conclusions about the properties of functions
Maximum property of odd functions
Symmetry and periodicity combined
The axis of symmetry is symmetric about the axis of symmetry to form a new axis of symmetry
The symmetry point is symmetric about the symmetry point to form a new symmetry point
The symmetry point is symmetrical about the symmetry axis to form a new symmetry point, and the symmetry axis is symmetrical about the symmetry point to form a new symmetry axis
Double symmetry produces periodicity. The period of the same symmetry (points and points or lines and lines) is 2 times the interval, and the period of different symmetries (points and lines) is 4 times the interval.
Properties of functions that are symmetric about points
graph of function
Point drawing method
Determine function domain
Simplify function analytic expression
Talk about properties of functions
Parity
Monotonicity
cyclical
symmetry
list
Draw points
Connect
Image transformation method for drawing
translation transformation
Pan left and right
Pan up and down
Add left and subtract right, add up and subtract
Scaling transformation
Horizontal scaling
Longitudinal telescopic
Symmetric transformation
Symmetric about the x-axis
Symmetric about the y-axis
Symmetric about the origin
flip transformation
Fold left and right
The part on the right side of the y-axis is folded to the left, the part on the left side of the original y-axis is removed, and the right side remains unchanged.
Fold up and down
The lower part of the x-axis is folded to the upper part, and the x-axis and the upper part remain unchanged.
Functions and Equations
zero point of function
concept
The relationship between the zero points of the function, the intersection of the graph of the function and the x-axis, and the roots of the corresponding equations
Theorem about the existence of function zeros
The existence theorem of zero points can only determine whether there are zero points in the open interval, but cannot determine the number of zero points.
Common conclusions
If a periodic function has zero points, there must be infinitely many zero points.
inverse functions of each other
piecewise function
Domain
The union of the definition domains of each segment of functions
range
The union of the value ranges of each segment function
Correspondence
Functions have different correspondences on different subsets of their domain
Can be one-to-one, many-to-one, not one-to-many
Non-empty number set B
Non-empty number set A
Domain of abstract function
The independent variable is the value range of x, and the value ranges in parentheses are equal
same function
①The definition domain is the same
②The corresponding relationship is the same
If the definition domain and corresponding relationship of the function are the same, the value range must be the same.