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 → Functions and Graphs ↓


Types of Functions


A function is a fundamental concept in algebra and calculus used to describe relationships between two sets. In mathematics, a function describes how each input (x-value) from one set is related to exactly one output (y-value) in another set. A function is often written as f(x) where f is the name of the function, and x is the input value or variable.

Linear functions

Linear functions are the simplest type of function where the graph is a straight line. The general form of a linear function is:

f(x) = mx + b

where m is the slope of the line, and b is the y-intercept, or the point where the line crosses the y-axis. For example, if we have a function:

f(x) = 2x + 3

The slope is 2, which means the line rises 2 units for every 1 unit moved to the right. The y-intercept is 3, which indicates the line crosses the y-axis at the point (0, 3).

XY(0, 3)

Examples of linear functions:

Some examples include:

f(x) = 5x - 1
f(x) = -3x + 4
f(x) = x

In each function, you can identify m and b and predict where the line will be based on these parameters.

Quadratic functions

The quadratic function includes terms where the variable is squared. Its general form is:

f(x) = ax^2 + bx + c

where a, b, and c are constants. Quadratic functions form a parabolic graph. For example, consider:

f(x) = x^2 + 2x + 1

In this function, the parabola opens upward and its vertex can be found using the formula for x = -b/(2a). For a, the parabola opens upward, resembling a "U" shape, and for a, it opens downward.

XY

Examples of quadratic functions:

Some quadratic functions include:

f(x) = 2x^2 + 3x + 5
f(x) = -x^2 + 4x - 7
f(x) = x^2

Cubic functions

Cubic functions are functions where the highest power of x is 3. The general form is:

f(x) = ax^3 + bx^2 + cx + d

Cubic functions have a variety of shapes, but they always involve a curve whose slope usually changes. For example:

f(x) = x^3 - 3x^2 + x

This shows a classic cubic behavior where the graph may have one or two inflection points.

XY

Examples of cubic functions:

Examples include:

f(x) = x^3 + 6x^2 - 15x + 5
f(x) = -2x^3 + x^2 - x + 7
f(x) = x^3

Exponential functions

Exponential functions have the variable in the exponent and they are of the form:

f(x) = a * b^x

Here, a is a constant, b is the base of the exponential function, and x is the exponent. These functions show rapid growth or decline depending on whether b is greater than 1 or between 0 and 1. For example:

f(x) = 2 * 3^x

Graphs of exponential functions will show a very rapid increase upward, or a decay if the function is decreasing.

XY

Examples of exponential functions:

Some typical examples are as follows:

f(x) = 5 * (1/2)^x
f(x) = 3^x
f(x) = 10 * 2^x

Logarithmic functions

The logarithmic function is the inverse of the exponential function and has the form:

f(x) = log_b(x)

The base b is a positive number that is not equal to 1. Logarithmic functions grow more slowly than exponential functions. For example:

f(x) = log_2(x)
XY

Examples of logarithmic functions:

Examples include:

f(x) = log_3(x)
f(x) = log_10(x)
f(x) = ln(x)

Trigonometric functions

Trigonometric functions are related to the angles of triangles and recurring events. Common trigonometric functions include sine, cosine, and tangent which are represented as follows:

f(x) = sin(x)
f(x) = cos(x)
f(x) = tan(x)

These functions have characteristic patterns, oscillating between values and often modeling cyclical behavior, like waves.

X

For example, a sine wave repeats every 360 degrees or 2π radians, and ranges in value from -1 to 1.

Examples of trigonometric functions:

Examples are:

f(x) = 3 * sin(x)
f(x) = 2 * cos(x)
f(x) = tan(x) + 1

Conclusion

Understanding the different types of functions provides a broad foundation for describing different types of relationships between variables. Each function serves unique purposes and exhibits special characteristics, which affect the behavior of their graphs. By identifying function types, you can predict how they behave and solve real-world problems involving patterns and data modeling.


Grade 11 → 2.1


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Types of Functions

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