Vertical and Horizontal Stretches in Function Transformations

Introduction to Vertical Stretches in Functional Notation

• Impact of Leading Factors: When a numerical factor is placed at the front of an equation, such as $y = 3\times\text{f}(x)$, it changes the visual representation of the graph significantly. In the example provided, the equation for a semicircle, $y = 3\times\text{f}(x)$ where $\text{f}(x) = \sqrt{4 - x^2}$, is explored.

• Definition of Vertical Stretch: The presence of the coefficient 3 results in a vertical stretch by a factor of 3 about the x-axis.

• Exam Terminology Warning: On formal exams, specifically the diploma exam, students must write out the full phrase "vertical stretch" rather than using abbreviations. This ensures the marker recognizes the specific transformation being described.

• The "Rubber Band" Metaphor: A vertical stretch can be visualized by imagining a graph pinned down to the x-axis. If you grab the graph and pull it vertically, it expands like a rubber band.

• The Meaning of "Factor": In mathematical terms, the word "factor" indicates product and multiplication. When describing a transformation, leaving out the word "factor" (e.g., "the graph is stretched 3") makes the description mathematically imprecise.

Mechanics and Calculation of Vertical Stretches

• Distance Analysis: The stretch factor is applied to the distance between each coordinate and the x-axis.

  • Example 1: If a coordinate is initially 2 units away from the x-axis (e.g., at a height of 2), a vertical stretch by a factor of 3 triples that distance, moving the coordinate to 6 units away ($2 \times 3 = 6$).

  • Example 2: A coordinate that was 1 unit away from the x-axis moves to 3 units away ($1 \times 3 = 3$).

• Invariance: Coordinates located directly on the x-axis have a distance of 0 from the axis. Since $0 \times 3 = 0$, these points do not change position.

  • The x-intercepts of a graph remains invariant (unchanged) under any vertical stretch.

• Impact on Y-Intercepts: For the initial semicircle graph with a y-intercept at 2, a vertical stretch factor of 3 triples the intercept to 6.

Vertical Compressions (Vertical Stretches Between 0 and 1)

• Vertical Stretch by a Factor of 1/2: If the equation is updated to $y = \frac{1}{2}\text{f}(x)$, the graph becomes shorter rather than taller.

• Terminology Variation: While some textbooks use the term compression (for factors between 0 and 1) or expansion (for factors greater than 1), the term "stretch" is universally applicable as long as the specific factor is included (e.g., stretch by a factor of $1/2$).

• Observations for Factor of 1/2:

  • The x-intercepts remain invariant.

  • The y-intercept is halved (reduced to 1 if the original was 2).

Theoretical Foundation: Transformation by Replacement

• The Replacement Principle: All functional transformations can be viewed as the result of a variable replacement.

• Vertical Stretch Replacement: If the equation is $y = a\text{f}(x)$, it can be rewritten as $\frac{1}{a}y = \text{f}(x)$. This implies that initial variable $y$ was replaced by $\frac{1}{a}y$.

• The Reciprocal Rule: The stretch factor and the replacement number are always reciprocals of each other.

  • If the vertical stretch factor is 3, the replacement for $y$ is $\frac{1}{3}y$.

  • If the vertical stretch factor is $\frac{1}{2}$, the replacement for $y$ is $2y$.

Horizontal Stretches and the Y-Axis

• Notation: Horizontal stretches occur when the variable within the function notation is modified, such as in $y = \text{f}(bx)$.

• Replacement and Factor Reciprocity: Just as with vertical transformations, the replacement value and the stretch factor are reciprocals.

  • Example (f(4x)): If $x$ is replaced by a factor of 4, the resulting transformation is a horizontal stretch by a factor of 1/4 about the y-axis.

• Equation Derivation: For the function $\text{f}(x) = \sqrt{4 - x^2}$, the substituted form $\text{f}(4x)$ becomes:   $y = \sqrt{4 - (4x)^2}$   $y = \sqrt{4 - 16x^2}$

• Visual Impact: A horizontal stretch by a factor of $1/4$ narrows the graph. An x-intercept at 2 would move to $2 \times \frac{1}{4} = 0.5$.

• Horizontal Expansions: If the equation is $y = \text{f}(\frac{1}{3}x)$, the horizontal stretch factor is 3. This widens the graph.

  • Example: An x-intercept at $-2$ moves out to $-6$, and positive 2 moves out to 6.

• Invariance: Under horizontal stretches, the y-intercept remains invariant because its distance to the y-axis is 0.

Summary of Transformation Rules

• Vertical Stretches ($y = a\text{f}(x)$):

  • Happens about the x-axis.

  • Stretch factor is $a$.

  • If 0 < a < 1, the graph undergoes compression (shorter).

  • If a > 1, the graph undergoes expansion (taller).

  • If $a$ is negative, there is a reflection in the x-axis in addition to the stretch.

• Horizontal Stretches ($y = \text{f}(bx)$):

  • Happens about the y-axis.

  • Stretch factor is $\frac{1}{b}$.

  • If 0 < b < 1, the stretch factor is greater than 1, resulting in expansion (wider).

  • If b > 1, the stretch factor is less than 1, resulting in compression (thinner).

  • If $b$ is negative, there is a reflection in the y-axis in addition to the stretch.

Step-by-Step Problem Solving and Examples

Example 1: Writing Replacements and Equations

• Task A: Horizontal stretch by factor of 6 about the y-axis.

  • Replacment: Replace $x$ with $\frac{1}{6}x$.

  • Equation: $y = \text{f}(\frac{1}{6}x)$.

• Task B: Vertical stretch by factor of 1/5 about the x-axis.

  • Replacement: Replace $y$ with $5y$.

  • Preliminary Equation: $5y = \text{f}(x)$.

  • Simplified Equation: $y = \frac{1}{5}\text{f}(x)$.

• Task C: Reflection in the x-axis and a vertical stretch by factor of 3.

  • Replacement 1 (Stretch): Replace $y$ with $\frac{1}{3}y$.

  • Replacement 2 (Reflection): Replace $y$ with $-y$

  • Combined replacement: Replace $y$ with $-\frac{1}{3}y$.

  • Simplify: $-\frac{1}{3}y = \text{f}(x) \rightarrow y = -3\text{f}(x)$.

Example 2: Describing Changes from Notation

• Change f(4x): Horizontal stretch, factor $1/4$, about the y-axis.

• Change f(x) with y replaced by (1/3)y: Vertical stretch, factor 3, about the x-axis.

• Change with constant y replaced by 6y and x by (1/3)x: Vertical stretch factor $1/6$ about the x-axis AND horizontal stretch factor 3 about the y-axis.

Example 3: Sketching a Graph with Combined Transformations

• Scenario: Sketching $y = \text{f}(-2x)$.

• Transformations:

  • Horizontal stretch by a factor of $1/2$ about the y-axis.

  • Reflection in the y-axis.

• Process: Take each coordinate, multiply its horizontal distance to the y-axis by $1/2$, then move it to the opposite side of the y-axis.

Advanced Algebraic Simplifications

Squaring Transformations

• Equation: $y = x^2$ with horizontal stretch factor $3/4$.

• Replacement: Replace $x$ with $\frac{4}{3}x$.

• Transformation: $y = (\frac{4}{3}x)^2 = \frac{16}{9}x^2$.

Radical Transformations

• Equation: $y = \sqrt{x} - 3$ with horizontal stretch factor 4 and vertical stretch factor 2.

• Horizontal replacement: $x \rightarrow \frac{1}{4}x$.

• Vertical replacement: $y \rightarrow \frac{1}{2}y$.

• Solving for $y$:   $\frac{1}{2}y = \sqrt{\frac{1}{4}x} - 3$   $\frac{1}{2}y = \frac{1}{2}\sqrt{x} - 3$   $y = 2 \times (\frac{1}{2}\sqrt{x} - 3)$   $y = \sqrt{x} - 6$

Linear Transformations

• Equation: $y = 3x + 7$ with vertical stretch factor $1/3$.

• Replacement: Replace $y$ with $3y$.

• Transformation: $3y = 3x + 7 \rightarrow y = x + \frac{7}{3}$.

Practical Observation from Graphs

• Identifying Transformations: When looking at a thin original line and a thick transformed line, compare key points like peaks or intercepts.

• Vertical Case: If the original height is 6 and the new height is 3, the vertical stretch factor is $1/2$.

• Equation Derivation from Observation: Starting from $y = \frac{6}{x^2 + 1}$, applying a vertical stretch factor of $1/2$ results in the replacement of $y$ with $2y$.   $2y = \frac{6}{x^2 + 1} \rightarrow y = \frac{3}{x^2 + 1}$.

Classroom Logistics

• Instructional Note: Practice exercises are essential for mastering reciprocity between stretch factors and replacement values.

• Exam Security: Cell phones must be stored away and are strictly prohibited during the return of unit exams; all mobile devices must stay in the classroom and cannot be taken home while exams are being reviewed.