Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 ↓


Algebra


Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In algebra, those symbols (and their arithmetic operations) are used to represent real-world situations.

Basic concepts of algebra

Variables

In algebra, a variable is a symbol used to represent a number. It is usually a letter such as x, y, or z. For example, in the equation x + 2 = 5, x is a variable.

Constants

Constants are fixed values. They do not change. Examples of constants include numbers such as 3, -7 and 10.5.

Expression

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. An example of an algebraic expression is 4x + 7.

Equation

An algebraic equation is a statement that two algebraic expressions are equal. For example, 3x + 2 = 11 is an equation. Equations can have one or more solutions.

x = 3 Solution

Operations in algebra

Addition and subtraction

Adding or subtracting algebraic terms involves combining like terms. Like terms are terms that have variables raised to the same power. For example:

4x + 3x = 7x
5a - 2a = 3a

Multiplication

When multiplying terms, you multiply the coefficients and add the exponents of the same bases. For example:

2x * 3x = 6x^2
4a * 5b = 20ab

Division

When dividing terms, you divide the coefficients and subtract the exponents of the same bases. For example:

8x^2 / 2x = 4x
15a^3 / 5a = 3a^2

Solving linear equations

Solving equations means finding the value of the variable that makes the equation true. Here is the basic step-by-step method for solving linear equations:

Example 1

Solve the equation 2x + 3 = 11.

Step 1: Subtract 3 from both sides.
2x + 3 - 3 = 11 - 3
2x = 8

Step 2: Divide both sides by 2.
2x / 2 = 8 / 2
x = 4

Example 2

Solve the equation 5y - 7 = 18.

Step 1: Add 7 to both sides.
5y - 7 + 7 = 18 + 7
5y = 25

Step 2: Divide both sides by 5.
5y / 5 = 25 / 5
y = 5

Graphs of equations

The graph of an equation is the set of all points that satisfy the equation. For linear equations, this graph is a straight line. For example, the graph of y = 2x + 3 is a straight line.

y = 2x + 3

Polynomials

A polynomial is an expression consisting of variables and coefficients, built using addition, subtraction, multiplication, and non-negative integer exponents of the variables.

Example polynomials

An example of a polynomial is 3x^2 + 2x - 5. This polynomial has three terms:

  • 3x^2 is called a quadratic term.
  • 2x is called the linear term.
  • -5 is called the constant term.

Factorization of polynomials

Factoring is the process of breaking down a complex expression into simpler factors that can be multiplied to get the original expression back. For example, factoring the polynomial x^2 - 5x + 6 gives:

x^2 - 5x + 6 = (x - 2)(x - 3)

Quadratic equations

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. To solve a quadratic equation, you can use:

  • Factoring
  • Completing the square
  • Quadratic formula

Quadratic formula

The quadratic formula to solve ax^2 + bx + c = 0 is:

x = (-b ± √(b^2 - 4ac)) / (2a)

Example

Solve the equation 2x^2 + 3x - 2 = 0 using the quadratic formula:

a = 2, b = 3, c = -2
Step 1: Plug these into the quadratic formula:
x = (-3 ± √(3^2 - 4 * 2 * -2)) / (2 * 2)
Step 2: Simplify under the square root:
x = (-3 ± √(9 + 16)) / 4
x = (-3 ± √25) / 4
Step 3: Solve for x:
x = (-3 ± 5) / 4
The solutions are:
x = (2) / 4 = 0.5
x = (-8) / 4 = -2

Inequality

Inequalities are mathematical statements that relate expressions that are not necessarily equal. They are usually written with one of the symbols <, ≤, >, ≥. For example, solving the inequality x + 3 > 5 is the same as solving an equation.

Example

Solve the inequality 2x - 3 < 7.

Step 1: Add 3 to both sides.
2x - 3 + 3 < 7 + 3
2x < 10

Step 2: Divide both sides by 2.
2x/2 < 10/2
x < 5

Systems of linear equations

A system of linear equations is a set of two or more equations that have the same variables. The solution of a system of equations is a set of values of the variables that satisfy each equation in the system.

Solution methods

There are several ways to solve systems of linear equations:

  • Graph
  • Replacement
  • Elimination

Example (substitution)

Solve the system of equations:

1. x + y = 10
2. 2x - y = 1

Step 1: Solve Equation 1 for y:

y = 10 - x

Step 2: Substitute y in Equation 2:

2x - (10 - x) = 1

Simplify x and solve:

2x - 10 + x = 1
3x = 11
x = 11 / 3

Step 3: Substitute the value of x into Equation 1 to get y:

x + y = 10
11/3 + y = 10
y = 10 - 11/3
y = 30/3 - 11/3
y = 19/3

Conclusion

Algebra serves as a strong foundation for higher-level mathematics and provides techniques for solving a variety of mathematical problems. Through understanding and practicing algebraic concepts such as variables, equations, polynomials, and systems of equations, students can develop strong problem-solving skills that are applicable in both academics and the real world.


Grade 11 → 1


U
username
0%
completed in Grade 11

Next (1.1) →
Sequences and Series

Comments 



Algebra

https://www.buddymath.com/g11/1?bmal=en