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 ↓


Polynomial Functions


Polynomial functions are a class of mathematical functions that are algebraic expressions. Understanding polynomial functions is essential because they form the basic structure of various mathematical modeling processes. They appear in many real-world scenarios and are the building blocks of many more complex functions found in mathematics.

What are polynomial functions?

A polynomial function is a function that can be represented in the following general form:

f(x) = a n x n + a n-1 x n-1 + ... + a 1 x + a 0

Here, a n , a n-1 , ..., a 0 are constants known as the coefficients of the polynomial, and n is a non-negative integer known as the degree of the polynomial. The highest power of x (which is n in a n x n) is the degree of the polynomial. The term a n must not be zero for n to be a degree.

Types of polynomial functions

Polynomial functions come in different forms depending on their degree:

  • Stable functions:
    A polynomial function with degree 0. For example, f(x) = 7 The graph of a constant function is a horizontal line. 
    f(x) = c
    f(x) = c
  • Linear function:
    A polynomial function with degree 1. For example, f(x) = 2x + 3 Linear functions graph as straight lines. 
    f(x) = mx + c
    f(x) = mx + c
  • Quadratic function:
    A polynomial function with degree 2. For example, f(x) = x 2 - 4x + 4 graphs a quadratic as a parabola. 
    f(x) = ax 2 + bx + c
    f(x) = x 2 + bx + c
  • Cube function:
    A polynomial function with degree 3. For example, f(x) = x 3 - 3x 2 + x - 2 Cubic functions show more complexities and many twists. 
    f(x) = ax 3 + bx 2 + cx + d
    f(x) = x 3 + bx 2 + cx + d
  • Quartile function:
    A polynomial with degree 4, such as f(x) = x 4 - 2x 2 + 1 can produce graphs that appear wavy or U-shaped. 
    f(x) = ax 4 + bx 3 + cx 2 + dx + e
    f(x) = ax 4 + bx 3 + cx 2 + dx + e

Graphing polynomial functions

Graphing polynomial functions helps to see the roots and behaviour of the function. Some important features to note while plotting the graph are as follows:

  • Interception:
    • Y-intercept: The point where the graph crosses the y-axis. Found by evaluating f(0).
    • X-intercepts (origins): The points where the graph crosses or touches the x-axis, found by solving f(x) = 0.
  • End Behavior:
    • Determined by the leading term (highest power term) of the polynomial.
    • If the degree is even, both ends will go in the same direction, up if the leading coefficient is positive, and down if it is negative.
    • If the degree is odd, the ends move in opposite directions, shifting up to the right for a positive leading coefficient and down to the right for a negative leading coefficient.
  • Turning Point:
    • The maximum number of turning points is n-1, where n is the degree of the polynomial.

Examples of polynomial functions

Let's look at some examples and learn how to identify the degree, leading coefficient, and graph a polynomial:

Example 1: Quadratic polynomial

Given: f(x) = x 2 - 3x + 2

  • Degree: 2 (quadratic)
  • Leading coefficient: 1
  • Y-intercept: f(0) = 2
  • X-intercept: Solving for x 2 - 3x + 2 = 0 gives the roots x = 1 and x = 2
f(x) = x 2 - 3x + 2

Example 2: Cubic polynomial

Given: f(x) = x 3 - 4x

  • Degree: 3 (cubic)
  • Leading coefficient: 1
  • Y-intercept: f(0) = 0
  • X-intercept: Solving for x 3 - 4x = 0 gives roots at x = 0, x = 2, and x = -2
f(x) = x 3 - 4x

Properties of polynomial functions

When dealing with polynomial functions, there are several inherent properties:

  • Continuous and Smooth: Polynomial functions are continuous everywhere. This means that the graph has no breaks, jumps, or sharp edges.
  • Factorable: any polynomial can be factored into linear factors (may involve complex or imaginary numbers for some roots).
  • Maximum degree: The number of x-intercepts and local maximums/minimums is finite compared to the degree.

Applications of polynomial functions

Polynomial functions are widely used in various fields:

  • Physics: Polynomial functions represent the motion of objects where the acceleration remains constant.
  • Economics: Used to prepare cost, revenue, and profit models.
  • Engineering: modeling of loads and forces, as well as the analysis of structures and systems.
  • Biology: Study of population growth rates and various biological systems.

Conclusion

Polynomial functions are fundamental in mathematics and science. Understanding their structure, types, graphing methods, and applications not only enhances problem-solving skills but also provides a foundation for studying more complex equations and functions in advanced mathematics. Whether you are predicting the behavior of a physical system or analyzing complex data trends, polynomial functions serve as invaluable tools in your mathematical toolkit.


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Polynomial Functions

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