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Basic knowledge of high school mathematics (quadratic functions, equations and inequalities)
This is a mind map about the basic knowledge of high school mathematics (quadratic functions, equations and inequalities), including the properties of equality and inequality, Basic inequalities (mean inequality), etc.
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Basic knowledge of high school mathematics (quadratic functions, equations and inequalities)
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Quadratic functions, equations and inequalities of one variable
Quadratic functions, quadratic equations, and inequalities
Quadratic function
graph of quadratic function
The relationship between the graph of function y=x² and function y=ax² (a≠0)
The image of y=ax² (a≠0) is obtained by keeping the abscissa coordinate of each point of the image of y=x² unchanged and the ordinate becoming a times the original value.
a determines the direction and size of the image opening. The larger a is, the smaller the image opening is.
The relationship between the graph of function y=ax² (a≠0) and function y=a(x h)² k (a≠0)
y=ax² passes through {h>0, translate h unit length to the left; h<0 translate h unit length to the right} to get y=a(x h)²
y=a(x h)² passes through {k>0, translate upward k unit lengths; k<0, translate downward k unit lengths} to get y=a(x h)² k
After the function y=ax² bx c (a≠0) is formulated into the form of y=a(x h)² k, it is obtained by shifting the image of y=ax² (a≠0) left and right
Properties of Quadratic Functions
Three properties of quadratic functions
If the vertex coordinates of the quadratic function (-h, k) are known, the quadratic function can be expressed as y=a(x h)² k (a≠0)
If the two roots of the equation ax² bx c=0 (a≠0) are known to be x1 and x2 (the intersection of the parabola and the X-axis of the abscissa), then the quadratic function can be expressed as y=a(x-x1)(x -x2)（a≠0）
Properties of function y=ax² bx c (a≠0)
Function a>0
opening direction
up
Vertex coordinates
(-b/(2a),(4ac-b²)/(4a))
Symmetry axis
x=-b/(2a)
Maximum and minimum value problems
When x=-b/(2a), the function has a minimum value (4ac-b²)/(4a); there is no maximum value
Function a<0
opening direction
down
Vertex coordinates
(-b/(2a),(4ac-b²)/(4a))
Symmetry axis
x=-b/(2a)
Maximum and minimum value problems
When x=-b/(2a), the function has a maximum value (4ac-b²)/(4a); there is no minimum value
The concept of quadratic equation of one variable
concept
An equation where both sides of the equal sign are integers, contain only one unknown (unary), and the highest degree of the unknown is quadratic.
General form: y=ax² bx c (a≠0)
Solution of Quadratic Equation
Also called the roots of a quadratic equation of one variable
1. When a≠0, it can be said that the equation is a quadratic equation. 2. If the text clearly states that y=ax² bx c is a quadratic equation, it implies the condition of a≠0 3. c is a constant term (or can be regarded as a coefficient of a zero-order term)
Solution to quadratic equation of one variable
Solve quadratic equations of one variable using direct square root
Generally, the method of using the definition of square root to directly take the square root to find the solution of a quadratic equation is called the direct square root method.
For a quadratic equation of the form (ax b)²=c (c≥0), the solution is x=(±Ö(c) -b)/a
Note: when using the direct square root method, c ≥ 0, and when taking the square root, pay attention to ±√c
Solving quadratic equations of one variable using formula method
definition
A quadratic equation of the form ax² bx c=0 (a≠0) is transformed into a completely square equation with an unknown number on the left end, and a non-negative constant on the right end, which can be solved directly by the square root method.
General steps
Move item
Make the left-hand side of the equation contain only quadratic terms and linear terms, and the right-hand side be constant terms
Set a to 1
Divide both sides of the equation by the coefficient of the quadratic term to change the coefficient of the quadratic term to 1
formula
Add half the square of the coefficient of the linear term to both sides of the equation (i.e. add [b/(2a)]² in the general form) Convert the original equation into the form of (x-n)²=m (that is, converted into: [x b/(2a)]²=(b²-4ac)/(4a²))
If m≥0, then directly use the square root method to solve
If m<0, then the original equation has no real roots, that is, the equation has no real solutions.
Solving quadratic equations of one variable using formula method
In ax² bx c=0 (a≠0), when b²-4ac≥0, put a, b, c into the formula x=[-b±Ö(b²-4ac)]/(2a) to get the equation of root
The derivation process of the root formula of a quadratic equation is obtained by following the square root shift of the general steps in the coordination method.
The premise of using the formula method to solve a quadratic equation of one variable is b²-4ac≥0, where Δ=b²-4ac is called the discriminant
If Δ=b²-4ac>0, then the equation has two different real roots, x=[-b±Ö(b²-4ac)]/(2a)
If Δ=b²-4ac=0, then the equation has two identical real roots, x1=x2=-b/(2a)
If Δ=b²-4ac<0, then there are no real roots
The role of Δ=b²-4ac 1. Determining the roots without solving the equation 2. Determine the value range of the letter coefficient according to the equation 3. Discuss and solve problems related to the roots of quadratic equations of one variable 4.Δ=0 means that the equation has two identical roots instead of only one root.
General steps for solving quadratic equations of one variable using the formula method
Transform into the general form ax² bx c=0 (a≠0)
Determine the values of a, b, c
Calculate the value of Δ=b²-4ac
Determine the root situation based on the value of Δ=b²-4ac
If there are real roots, use the formula method to solve the equation x=[-b±Ö(b²-4ac)]/(2a)
1. When the equation contains unknown letters, it needs to be regarded as a constant. First, organize the equation into the general form of an equation about the unknown, and then use the root finding formula under the premise that b²-4ac≥0 2. Pay attention to the value range of the letters in the question and discuss it.
Factoring method to solve quadratic equations of one variable
Definition of factoring
definition
When solving a quadratic equation, first factor it, so that the equation becomes a form in which the product of two linear equations is equal to 0, and then make the two linear equations equal to 0 respectively, thereby achieving degree reduction. This kind of solving a quadratic equation The method is called factorization method
Theoretical basis
The product of two factors is equal to zero, then at least one of the two factors is equal to zero, that is, if ab=0, then a=0, or b=0
main method
Extract common factors method
Use the squared difference formula
a²-b²=(a b)(a-b)
Use the perfect square formula
a²±2ab b²=(a±b)²
cross multiplication
If in x² Cx D=0, we can find D=ab, C=a b, then x² Cx D=(x a)(x b)
The relationship between the roots and coefficients of a quadratic equation
Relationship between roots and coefficients
Vedic theorem
x1 x2=-b/a, x1·x2=c/a
Important Corollary to the Relationship between Roots and Coefficients
Corollary 1
If the equation x² px q=0, then x1 x2=-p, x1·x2=q
Corollary 2
A quadratic equation of one variable with two numbers x1 and x2 as roots (the coefficient of the quadratic term is 1) can be expressed as: x²-(x1 x2)x x1·x2=0
Included conditions
The equation is a quadratic equation of one variable, that is, the coefficient of the quadratic term is not zero, a≠0
The equation has real roots, that is, if Δ=b²-4ac≥0
corollary variant
x1² x2²=(x1² 2x1·x2 x2²)-2x1·x2=(x1 x2)²-2x1·x2
1/x1 1/x2=(x1 x2)/(x1·x2)
(x1 a)(x2 a)=x1·x2 a(x1 x2) a²
|x1-x2|=√((x1-x2)²)=√((x1 x2)²-4x1·x2)
Discuss the relationship between roots and coefficients. Symbols of roots
If the two roots of the quadratic equation ax² bx c=0 (a≠0) are x1 and x2, then
Δ≥0, and x1·x2>0
x1 x2>0
Both roots are positive numbers
x1 x2＜0
Both are negative numbers
Δ＞0, and x1·x2＜0
x1 x2>0
The two roots have different signs and the positive root has a larger absolute value.
x1 x2＜0
The two roots have different signs and the negative root has a larger absolute value.
Solutions to equations and systems of equations
Generally, the set of all solution combinations of an equation is called the solution set of this equation.
The intersection of the solution sets of each equation is the solution set of the system of equations.
Quadratic inequality of one variable
concept
definition
Generally, we call an inequality that contains only one unknown number and the highest degree of the unknown number is 2, called a quadratic inequality of one variable. The general form of a quadratic inequality of one variable is ax² bx c＞0, or ax² bx c＜0, where a, b, c are all constants, a≠0
Expression, where a, b, c are all constants, a≠0
ax² bx c≤0
ax² bx c＜0
ax² bx c≥0
ax² bx c>0
Solution set, where a, b, c are all constants, a≠0
ax² bx c≥0
The set of values of the independent variable x such that the function value of y=ax² bx c is greater than or equal to 0
ax² bx c>0
The set of values of the independent variable x such that the function value of y=ax² bx c is a positive number
ax² bx c≤0
The set of values of the independent variable x such that the function value of y=ax² bx c is less than or equal to 0
ax² bx c＜0
The set of values of the independent variable x such that the function value of y=ax² bx c is a negative number
zero point of quadratic function
Generally, for the quadratic function y=ax² bx c, we call the real number x that makes ax² bx c=0 the zero point of y=ax² bx c
Solution to Quadratic Inequality of One Variable
Δ=b²-4ac
Δ=b²-4ac>0
Δ=b²-4ac=0
Δ=b²-4ac＜0
y=ax² bx c
y=ax² bx c>0
y=ax² bx c=0
y=ax² bx c＜0
Combine the inequality relationship between the discriminant and the function, and solve the solution of the inequality through image analysis
Solutions to Fractional Inequalities
4 forms and solutions of fractional inequalities
f(x)/g(x)＞0 ⇔ f(x)·g(x)＞0
f(x)/g(x)＜0 ⇔ f(x)·g(x)＜0
f(x)/g(x)≥0 ⇔ f(x)·g(x)≥0, and g(x)≠0 ⇔ f(x)·g(x)＞0, and f(x)= 0
f(x)/g(x)≤0 ⇔ f(x)·g(x)≤0, and g(x)≠0 ⇔ f(x)·g(x)＜0, and f(x)= 0
The same solution relationship between inequalities and inequality groups
f(x)·g(x)≥0
f(x)≥0, and g(x)≥0
Or f(x)≤0, and g(x)≤0
f(x)·g(x)≤0
f(x)≥0, and g(x)≤0
Or f(x)≤0, and g(x)≥0
The problem of constant establishment of inequalities
The condition that the solution set of the inequality is R (or always true)
y=ax² bx c
if a=0
b=0，c>0
y=ax² bx c>0 is always true
b=0，c＜0
y=ax² bx c＜0 is always true
If a≠0
a>0, Δ<0
y=ax² bx c>0 is always true
a<0, Δ<0
y=ax² bx c＜0 is always true
A method to find the parameter value range when the inequality is constant
y=f(x)≤a always holds ⇔ f(x)max≤a
y=f(x)≥a always holds ⇔ f(x)min≥a
Distribution of roots of quadratic equation of one variable
Prerequisites
Suppose the equation ax² bx c=0 (Δ＞0, a≠0) has two unequal roots x1, x2, and x1＜x2, the corresponding function is y=ax² bx c
Case 1: Compare the magnitude of two roots with 0, that is, compare the positive and negative conditions of the roots
a>0
x1＜x2＜0
①Δ＞0 ②Axis of symmetry-b/(2a) < 0 ③f(0)>0
①Δ＞0 ②Axis of symmetry-b/(2a) < 0 ③a·f(0)>0
0＜x1＜x2
①Δ＞0 ②Axis of symmetry-b/(2a) > 0 ③f(0)>0
①Δ＞0 ②Axis of symmetry-b/(2a) > 0 ③a·f(0)>0
x1＜0＜x2
①f(0)＜0
①a·f(0)＜0
a<0
x1＜x2＜0
①Δ＞0 ②Axis of symmetry-b/(2a) < 0 ③f(0)＜0
①Δ＞0 ②Axis of symmetry-b/(2a) < 0 ③a·f(0)>0
0＜x1＜x2
①Δ＞0 ②Axis of symmetry-b/(2a) > 0 ③f(0)＜0
①Δ＞0 ②Axis of symmetry-b/(2a) > 0 ③a·f(0)>0
x1＜0＜x2
①f(0)>0
①a·f(0)＜0
Situation 2: Comparison of the sizes of two roots and k
a>0
x1＜x2＜k
①Δ＞0 ②Axis of symmetry-b/(2a) < k ③f(k)＞0
①Δ＞0 ②Axis of symmetry-b/(2a) < k ③a·f(k)＞0
k＜x1＜x2
①Δ＞0 ②Axis of symmetry-b/(2a) > k ③f(k)＞0
①Δ＞0 ②Axis of symmetry-b/(2a) > k ③a·f(k)＞0
x1＜k＜x2
①f(k)＜0
①a·f(k)＜0
a<0
x1＜x2＜k
①Δ＞0 ②Axis of symmetry-b/(2a) < k ③f(k)＜0
①Δ＞0 ②Axis of symmetry-b/(2a) < k ③a·f(k)＞0
k＜x1＜x2
①Δ＞0 ②Axis of symmetry-b/(2a) > k ③f(k)＜0
①Δ＞0 ②Axis of symmetry-b/(2a) > k ③a·f(k)＞0
x1＜k＜x2
①f(k)>0
①a·f(k)＜0
Case 3: Distribution of roots on the interval, where m＜n＜p＜q
a>0
m＜x1＜x2＜n
①Δ＞0 ②f(m)＞0 ③f(n)>0 ④m＜Axis of symmetry-b/(2a)＜n
①Δ＞0 ②f(m)·f(n)＞0 ③m＜Axis of symmetry-b/(2a)＜n
m＜x1＜n＜x2, or x1＜m＜x2＜n
①f(m)·f(n) < 0
①f(m)·f(n) < 0
m＜x1＜n＜p＜x2＜q
①f(m)＞0 ②f(n)＜0 ③f(p)＜0 ④f(q)＞0 or ①f(m)·f(n) < 0 ②f(p)·f(q) < 0
①f(m)·f(n) < 0 ②f(p)·f(q) < 0
a<0
m＜x1＜x2＜n
①Δ＞0 ②f(m)＜0 ③f(n)＜0 ④m＜Axis of symmetry-b/(2a)＜n
①Δ＞0 ②f(m)·f(n)＞0 ③m＜Axis of symmetry-b/(2a)＜n
m＜x1＜n＜x2, or x1＜m＜x2＜n
①f(m)·f(n) < 0
①f(m)·f(n) < 0
m＜x1＜n＜p＜x2＜q
①f(m)＜0 ②f(n)>0 ③f(p)>0 ④f(q)＜0 or ①f(m)·f(n) < 0 ②f(p)·f(q) < 0
①f(m)·f(n) < 0 ②f(p)·f(q) < 0
Case 4: Distribution of roots on the interval, x1＜m, x2＞n
a>0
①f(m)＜0 ②f(n)＜0
a<0
①f(m)＞0 ②f(n)>0
Special case
i
If there is f(m)=0 or f(n)=0 in the given f(x) function interval (m,n), then f(m)·f(n)<0 is not satisfied. From f(m)=0, or f(n)=0, it is easy to know that m or n is one of the solutions to the equation, that is, the equation can be written in the form of ax² bx c=(x-m)·(Ax B), which means that there is A factor (x-m) [or (x-n)], you can find the other root of the equation, thereby determining whether it belongs to the interval (m, n), and find the value or range of the parameter
ii
The above situation 1, situation 2, situation 3 and situation 4 are all the results of the discussion when Δ＞0, ignoring the situation of Δ=0. When actually solving the problem, be sure to consider whether there is a condition that satisfies the condition when Δ=0 Parameter value
Basic Inequalities (Mean Inequality)
important inequalities
If a, b∈R,
Then a²≥0 (if and only if a=0, the equal sign is obtained)
|a|≥0, (the equal sign is obtained if and only when a=0)
(a-b)²≥0
a² b²≥2ab
[(a² b²)/2]≥[(a b)/2]²
(a b)²≥4ab
Obtain the equal sign if and only if a=b
basic inequalities
If a>0, b>0
Then: (2ab)/(a b)≤(ab)^(1/2)≤(a b)/2≤[(a² b²)/2]^(1/2)
Basic inequality: harmonic mean ≤ geometric mean ≤ arithmetic mean ≤ square mean
Memory: Adjust the number and calculate the formula
When finding the optimal value of basic inequalities, it is necessary to satisfy one positive, two definite and three equal
The sum of positive numbers is a constant value, then the product of positive numbers has the maximum value
The product of positive numbers is a constant value, then the sum of positive numbers has a minimum value
Extensions of basic inequalities
Arithmetic mean of three positive numbers - geometric mean inequality
If a, b, c∈R, then: (a b c)/3 ≥ (abc)^(1/3)
The equal sign holds if and only if a=b=c
Arithmetic mean of n positive numbers - geometric mean inequality
If A1, A2,...An∈R, then: (A1 A2... An)/n ≥ (A1·A2·...An)^(1/n)
Equality and inequality properties
Equality and Inequality
The concept of equation
An expression containing an equal sign is called an equation
The concept of inequality
Use mathematical symbols ≠ ＞ ＜ ≥ ≤ to connect two numbers or algebraic expressions to express the inequality between them. Expressions containing these inequality signs are called inequalities.
The concept of inequalities in the same direction and inequalities in opposite directions
Inequality in the same direction
If the left-hand side of two inequalities is greater (or less) than the right-hand side, the two inequalities are called inequalities in the same direction.
heterogeneous inequality
If the left-hand side of one inequality is greater than the right-hand side and the right-hand side of another inequality is greater than the left-hand side, the two inequalities are called opposite inequalities.
Commonly used inequality signs
Greater than ＞, less than ＜, greater than or equal to (at least, no less than) ≥, less than or equal to (at most, no more than) ≤
Difference method compares two real numbers (algebraic expressions)
a-b>0, then a>b
a-b<0, then a<b
a-b=0, then a=b
To compare any two real numbers, you only need to determine the relationship between their difference and 0.
Basic properties of equations
If a=b, then b=a
If a=b, b=c, then a=c
If a=b, then a±c=b±c
If a=b, then ac=bc
If a=b, then a/c=b/c (c≠0)
Extension: If a=b, then a^n=b^n (n∈N,N≥2)
Extension: If a=b>0, then a^(1/n)=b^(1/n) (n∈N,N≥2)
Properties of Inequalities
1Symmetry
a＞b⇔b＜a
Reversible
2 Transitivity
a>b, b>c⇒a>c
In the same direction
3 Additivity
a＞b⇔a c＞b c
Reversible
transfer rule
a b＞c⇔a＞c-b
Reversible
4 Multiplyability
a>b, and c>0⇒ac>bc a>b, and c<0⇒ac<bc
Pay attention to the situation of c>0 or c<0
5 Additivity in the same direction
a>b, and c>d, ⇒a c>b d
Can be added in the same direction
6 Multiplicability in the same direction and in the same positive direction
a>b>0, and c>d>0, ⇒ac>bd
Same direction and same direction can be multiplied
7 exponentiability
a＞b＞0，⇒a^n＞b^n（n∈N,N≥2）
Tongzheng can be exponentiated
Inequalities in the same direction cannot be subtracted, and inequalities in opposite directions cannot be added.
Commonly used inequalities
reciprocal property
a>b, ab>0, ⇒(1/a)<(1/b)
Inequality property 4
a＜0＜b，⇒(1/a)＜(1/b)
a>b>0, and 0<c<d, ⇒(a/c)>(b/d)
0＜a＜x＜b (or a＜x＜b＜0), ⇒(1/b)＜(1/x)＜(1/a)
Fractional properties
If a>b>0, m>0, then
Properties of proper fractions
(b/a)<[(b m)/(a m)]
(b/a)>[(b-m)/(a-m)], where b-m>0
That is: if the same positive number is added to the numerator and denominator of a proper fraction at the same time, the value of the fraction becomes larger.
Improper fraction properties
(a/b)>[(a m)/(b m)]
(a/b)<[(a-m)/(b-m)], where b-m>0
That is: if the same positive number is added to the numerator and denominator of an improper fraction at the same time, the value of the fraction becomes smaller.