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 ↓


Dilations and Stretches


In mathematics, particularly functions and graphs, transformations are operations that change the general shape or position of a graph. These transformations include dilation and stretching, which are important for understanding how graphs can be manipulated. This guide will explore the concepts of dilation and stretching, giving you a deeper understanding of how they work, how they differ, and how they can be applied to functions. We will dive deep into both vertical and horizontal transformations and support these explanations with examples and visual aids.

Understanding the basics

A function can often be represented graphically as a curve on a coordinate grid. Transformations modify this curve. Types of transformations that affect the shape of the graph are known as stretching and extension.

In simple terms:

  • Scaling refers to expanding or compressing a graph by a scale factor, affecting its width, height, or both.
  • Stretching is a special case of dilation where either the vertical or the horizontal dimension is changed while the other dimension is kept constant.

The main difference between dilation and stretch is that dilation can affect both dimensions of the graph equally, while stretch modifies the graph only in a specific dimension.

Vertical stretch and pull

When we refer to vertical stretching and dilation, we are talking about the change in the y-coordinate while leaving the x-coordinate unchanged. Considering the basis function f(x), the equation for the vertical transformation is given as:

f(x) → a * f(x)

Here, a is a constant that acts as a scale factor. Let's break it down:

  • If a > 1, the graph of f(x) is stretched vertically. This means that the y-values of all points are multiplied by a, making the graph longer.
  • If 0 < a < 1, the graph is compressed vertically, causing all y-values to decrease, making the graph smaller.
  • If a = 1, then no change occurs, and the graph remains unchanged.
  • If a = -1, the graph is reflected across the x-axis.

Example

Consider the original function f(x) = x^2, which is a simple parabola. Applying vertical dilation with a = 3 gives the function:

f(x) = 3 * x^2

This transformation stretches the parabola, making it narrower. Conversely, using a = 0.5 will compress the graph:

f(x) = 0.5 * x^2

Visually, the parabola appears wider when the y-value is halved.

y = x^2 y = 3x^2

Horizontal stretching and extension

While vertical changes change the y-values, horizontal changes affect the x-values. The general formula is:

f(x) → f(b * x)

The constant b determines the extent of the horizontal transformation:

  • If b > 1, the graph is compressed horizontally. This means that the x-values are decreased, and the graph appears compressed inward.
  • If 0 < b < 1, then multiplying the x-values by a factor greater than one extends the graph horizontally.
  • If b = 1, the graph remains unchanged.

Example

Taking the same basis function f(x) = x^2 and applying horizontal expansion with b = 2, we get:

f(x) = (0.5x)^2 = 0.25x^2

The result is a horizontally stretched parabola. Conversely, using b = 0.5 compresses it:

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

Visually, the parabola appears narrower because x is squared when divided by 2.

y = x^2 y = (0.5x)^2

Combined conversion

The transformations are often not singular in nature. A graph can be subjected to both vertical and horizontal transformations simultaneously. By adjusting both scale factors, a for vertical and b for horizontal, we can obtain a combination of stretching and dilation:

f(x) → a * f(b * x)

Using this combined transformation, several aspects of the graph can be changed at the same time. It is important to remember the sequence of operations:

  • First, apply the horizontal transformation.
  • Then proceed with the vertical transformation.

Example

Applying the given function f(x) = x^2, a = 3 and b = 0.5 gives:

f(x) = 3 * (0.5x)^2 = 3 * 0.25x^2 = 0.75x^2

This combination results in a graph that is stretched vertically and compressed horizontally.

Practical implications and uses

Understanding dispersion and dispersion has many practical implications, especially in engineering and physics. Graph transformations can be used to model real-world phenomena or fine-tune a function to align with observed data. Consider their use:

  • Modeling of mechanical systems and their responses.
  • Graphics design, where images can be re-sized for different screens or print media.
  • Data analysis, where transformations help generalize results.

Conclusion

Stretching and stretching are powerful tools in the mathematician's toolkit, providing a wide variety of ways to transform and understand functions graphically. Through consistent practice and application, these transformations become intuitive operations, making more sophisticated graph adjustments possible and providing greater insight into the behavior of various functions.

Like many mathematical concepts, the key to mastering dilation and stretching is practice. Working with different functions, applying various transformations, and visualizing the results can deepen this understanding. Practice combined with theoretical knowledge ensures that these concepts become second nature.


Grade 11 → 2.2.3


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