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 → Transformations of Functions ↓


Translations


In the study of functions and graphs, it is important to understand transformations to understand the behavior of graphs. An important type of transformation is a translation. Translations are essentially shifts of the graph in the coordinate plane. They do not change the shape or orientation of the graph; rather, they simply move it to a different location.

What is translation?

Translation in mathematics means to move a figure, line, or graph without rotating it, resizing it, or altering it. When we translate a graph, we slide it horizontally, vertically, or both. In function notation, translation involves adding or subtracting constants to the function or its variables.

Types of translation

There are two main types of translation:

  • Horizontal translation: Moving the graph to the left or right.
  • Vertical translation: Moving the graph up or down.

Horizontal translation

Horizontal translation involves shifting the graph to the right or left by a certain number of units. The function f(x) is translated horizontally by changing the variable to f(x - k), where k is the number of units by which the graph is shifted.

  • If k > 0, the graph moves k units to the right.
  • If k < 0, the graph moves |k| units to the left.

Example of horizontal translation

Consider the function f(x) = x^2, which is a basic quadratic function that forms a parabola with its vertex at the origin (0,0).

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

This is the translation of the function f(x) = x^2 by 3 units to the right.

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

Vertical translation

Vertical translation involves moving the graph up or down by a certain number of units. The function f(x) is translated vertically by adding a constant c to the entire function: f(x) + c.

  • If c > 0, the graph moves up c units.
  • If c < 0, the graph moves down |c| units.

Example of vertical translation

Consider the function f(x) = x^2.

 f(x) = x^2 + 4

This function results in a transformation f(x) = x^2 4 units upward.

f(x) = x^2 + 4 f(x) = x^2

Combining translations

Translations can be combined to move the graph both horizontally and vertically. For example, you could have a function that undergoes both types of translations:

 f(x) = (x - h)^2 + k

In this case, h represents the horizontal shift, and k represents the vertical shift. The function f(x) = (x - h)^2 + k transforms the original function f(x) = x^2 to the right by h units and up by k units.

  • If h > 0, move h units to the right; if h < 0, move |h| units to the left.
  • If k > 0, move up k units; if k < 0, move down |k| units.

Example of a combined translation

Let's apply both horizontal and vertical translation to the function f(x) = x^2:

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

Here, the function is translated 2 units to the right and 3 units up.

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

Understanding the effects of translation

Translation in mathematics means understanding how each operation affects the state of the graph. Once you understand how translation works, it becomes easier to predict graphical changes and apply this knowledge to complex problems.

Practical examples and exercises

To make these concepts clear, let's consider some more examples and exercises:

Example 1

Consider this function:

 g(x) = (x + 1)^2 - 2

Determine the translation applied to f(x) = x^2:

  • The graph is shifted horizontally 1 unit to the left, because x + 1 represents motion in the opposite direction.
  • The graph is shifted vertically downward by 2 units because - 2 term decreases the output for each corresponding x value.

Example 2

Let's solve this translation:

 h(x) = 2x - 5

Pay attention here:

  • No horizontal translation exists because there is no addition or subtraction to the function argument affecting x.
  • The vertical translation is 5 units downward, which translates every point on the graph parallelly.

Key points

  • Translations only affect the position of the graph, not its size or shape.
  • Horizontal and vertical translation can be done simultaneously.
  • Translation is needed to build a knowledge graph and understand the tasks visually.

Interactive views

The next time you see a function modification, imagine how these constants move your graph on the plane. Understanding translations not only helps in graph drawing but also enhances your ability to predict function behaviors without calculations.

Conclusion

Once the use of positive and negative values for h (horizontal) and k (vertical) is understood, translating graphs becomes straightforward. This understanding provides insight into function modifications and how they appear graphically, which lays the groundwork for more advanced studies into transformations such as reflections and rotations in mathematical analysis.


Grade 11 → 2.2.1


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