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


Functions and graphs are fundamental concepts in mathematics that allow us to understand and visualize relationships between variables. These concepts are essential for solving problems in a variety of fields, including science, engineering, and economics. In this lesson, we will explore the basics of functions and graphs, discuss the different types of functions, and learn how to interpret and draw their graphs.

What is the function?

A function is a relationship between two sets of values, where each input value from the first set relates to exactly one output value in the second set. Think of it as a machine: you give it an input, and it gives you an output.

Mathematical representation of functions

Mathematically, a function can be represented using a rule or equation. Here's an example:

    f(x) = x + 2

In this function, f(x) is the output when x is the input. For each value you choose for x, the function gives you the output by adding 2 to x.

Types of tasks

There are many types of functions you may encounter. Here are some common functions:

Linear functions

Linear functions form straight lines when graphed. Their format is as follows:

    f(x) = mx + c

Here, m is the slope, and c is the y-intercept. Slope represents the steepness of the line, and the y-intercept is where the line crosses the y-axis.

y = x

Quadratic functions

Quadratic functions form parabolas (U-shaped or inverted U-shaped graphs). Their format is as follows:

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

The letter a determines the direction of the parabola (if a is positive, then it opens upward, and if negative, then downward).

y = x 2

Exponential function

Exponential functions involve the variable in the exponent. These have the form:

    f(x) = a * b^x

These functions grow rapidly and form J-shaped curves. If b is greater than 1, the function models exponential growth; if b is between 0 and 1, it models exponential decay.

y = 2x

Understanding the graph of a function

Graphs are visual representations of functions, showing how the output value changes for different input values. Let's learn how to interpret these graphs.

Cartesian plane

Graphs are often drawn on the Cartesian plane, which is made up of two number lines: the horizontal x-axis and the vertical y-axis. The point where they intersect is called the origin, labeled as (0, 0).

0 X Y

Plotting points

Each point on a graph can be described by a pair of numbers called coordinates, written as (x, y). To plot a point, you start at the origin, move along the x-axis by the first number, and then move up or down along the y-axis by the second number.

For example, to plot the point (3, 2), you start at the origin, move 3 units to the right, and then move 2 units up. This takes you to the point (3, 2).

(3, 2)

Extracting function properties from a graph

By looking at the graph of a function we can determine various properties such as:

  • X-intercept: The point where the graph crosses the x-axis, i.e. the output y is zero.
  • Y-intercept: The point where the graph crosses the y-axis, that is, the input x is zero.
  • Slope: For linear functions, the slope of the graph represents the rate of change of the function.
  • Vertex of a quadrilateral: The highest or lowest point of the parabola, depending on whether it opens up or down.

Conversion of functions

The function can be shifted, stretched, compressed, or flipped. These changes are called transformations. Here are some basic transformations:

Vertical and horizontal shifts

Shifting a function up or down involves adding or subtracting a constant value from the function. Here's an example:

    f(x) = x^2 + 3

This will cause the original quadratic function f(x) = x^2 to be shifted up 3 units.

Some thoughts

Reflection flips the graph of a function across an axis. For example, reflecting across the x-axis results in the following transformations:

    f(x) = -x^2

This shows the original quadratic opening changing from an upward opening to now a downward opening.

Conclusion

Understanding functions and their graphs is a vital skill in mathematics. It allows us to see patterns, interpret data, and solve problems effectively. With practice, you will become more comfortable identifying different types of functions and recognizing their patterns on different graphs. As you progress, these concepts will serve as the foundation for more advanced mathematics.


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