Stretching and Compressing Graphs

Introduction

Key Concepts

Understanding Graph Transformations

Graph transformations involve altering the position, shape, or size of the basic graph of a function. Among these transformations, stretching and compressing are essential for manipulating the amplitude or width of graphs, allowing for a deeper comprehension of function behavior under different conditions.

Stretching and Compressing Vertically

Vertical stretching and compressing affect the graph of a function by changing its amplitude. A vertical stretch makes the graph taller, while a vertical compression makes it shorter.

• Vertical Stretch: To stretch a graph vertically by a factor of $k$, multiply the function by $k$ where $k > 1$. The new function becomes $y = k \cdot f(x)$. For example, if $f(x) = x^2$, then $y = 2x^2$ is a vertical stretch by a factor of 2.
• Vertical Compression: To compress a graph vertically by a factor of $k$, where $0 < k < 1$, multiply the function by $k$. The new function is $y = k \cdot f(x)$. For instance, $y = 0.5x^2$ compresses the graph of $f(x) = x^2$ vertically by a factor of 0.5.

Vertical Stretch and Compression: Impact on Function Behavior

Stretching and compressing vertically change how rapidly a function grows or shrinks. A vertically stretched graph indicates a faster growth rate, while a vertically compressed graph shows a slower growth rate.

• Example: Consider $f(x) = \sin(x)$. Stretching vertically by a factor of 3 transforms it to $y = 3\sin(x)$, increasing its amplitude from 1 to 3.
• Example: Compressing $f(x) = \sin(x)$ vertically by a factor of 0.5 results in $y = 0.5\sin(x)$, reducing its amplitude to 0.5.

Stretching and Compressing Horizontally

Horizontal stretching and compressing alter the width of a graph by transforming the input variable. A horizontal stretch makes the graph wider, while a horizontal compression makes it narrower.

• Horizontal Stretch: To stretch a graph horizontally by a factor of $k$, where $k > 1$, replace $x$ with $\frac{x}{k}$ in the function. The new function is $y = f\left(\frac{x}{k}\right)$. For example, stretching $f(x) = x^2$ by a factor of 2 results in $y = \left(\frac{x}{2}\right)^2$.
• Horizontal Compression: To compress a graph horizontally by a factor of $k$, where $0 < k < 1$, replace $x$ with $\frac{x}{k}$. The new function is $y = f\left(\frac{x}{k}\right)$. For instance, compressing $f(x) = x^2$ horizontally by a factor of 0.5 yields $y = \left(\frac{x}{0.5}\right)^2$ or $y = (2x)^2$.

Horizontal Stretch and Compression: Effects on Function Graph Width

These transformations influence the rate at which the function approaches its limits. A horizontally stretched graph indicates a slower approach, while a horizontally compressed graph indicates a faster approach.

• Example: $f(x) = \sqrt{x}$ stretched horizontally by a factor of 3 becomes $y = \sqrt{\frac{x}{3}}$, making the graph wider.
• Example: Compressing $f(x) = \sqrt{x}$ horizontally by a factor of 0.5 results in $y = \sqrt{2x}$, making the graph narrower.

Combined Transformations

Often, multiple transformations are applied to a single function to achieve a desired graph shape. Stretching and compressing can be combined with other transformations like translations and reflections.

• Example: For the function $y = -2f\left(\frac{x}{3}\right) + 4$, the graph is vertically stretched by a factor of 2, horizontally stretched by a factor of 3, reflected over the x-axis, and translated upward by 4 units.

Real-World Applications

Understanding stretching and compressing of graphs is essential in modeling various real-world phenomena, such as physics (motion equations), economics (supply and demand curves), and biology (population growth models).

• Physics: Modifying the amplitude of a wave function to represent different energy levels.
• Economics: Adjusting supply and demand curves to reflect changes in market conditions.
• Biology: Modeling population growth with adjusted growth rates.

Mathematical Representation and Formulas

The general form for stretching and compressing transformations can be expressed mathematically for any function $f(x)$.

• Vertical Stretch: $y = k \cdot f(x)$, where $k > 1$.
• Vertical Compression: $y = k \cdot f(x)$, where $0 < k < 1$.
• Horizontal Stretch: $y = f\left(\frac{x}{k}\right)$, where $k > 1$.
• Horizontal Compression: $y = f(kx)$, where $k > 1$.

Examples and Visual Illustrations

To solidify understanding, consider the function $f(x) = \cos(x)$. Applying various stretches and compressions alters its graph as follows:

• Vertical Stretch: $y = 3\cos(x)$ increases the amplitude from 1 to 3.
• Vertical Compression: $y = 0.5\cos(x)$ decreases the amplitude to 0.5.
• Horizontal Stretch: $y = \cos\left(\frac{x}{2}\right)$ stretches the period from $2\pi$ to $4\pi$.
• Horizontal Compression: $y = \cos(2x)$ compresses the period to $\pi$.

Visual graphs illustrating these transformations can greatly aid in understanding their effects.

Key Differences Between Stretching and Compressing

While both stretching and compressing alter the size of the graph, they do so in opposite directions and affect different aspects of the function.

• Vertical Transformations: Stretching increases amplitude; compressing decreases amplitude.
• Horizontal Transformations: Stretching widens the graph; compressing narrows the graph.

Impact on Function's Domain and Range

Stretching and compressing transformations can influence the domain and range of a function.

• Vertical Transformations: Affect the range by altering the maximum and minimum values.
• Horizontal Transformations: Affect the domain by changing the spread of input values.

Inverse Transformations

Understanding inverse transformations is crucial for reverting a graph to its original state.

• Inverse of Vertical Stretch: To invert a vertical stretch by factor $k$, apply a vertical compression by $\frac{1}{k}$.
• Inverse of Horizontal Stretch: To invert a horizontal stretch by factor $k$, apply a horizontal compression by $\frac{1}{k}$.

Common Mistakes and Misconceptions

Students often confuse vertical and horizontal transformations or misapply the stretch/compression factors.

• Mistake: Applying a horizontal stretch factor to vertical transformations.
• Misconception: Believing that a compression factor less than 1 for horizontal transformations is the same as vertical transformations.

Careful practice and visualization can help overcome these challenges.

Graphing Techniques

Efficient graphing techniques ensure accurate representation of transformed functions.

• Step 1: Identify the base function and its key features (intercepts, extrema, asymptotes).
• Step 2: Apply vertical transformations, adjusting the amplitude accordingly.
• Step 3: Apply horizontal transformations, altering the period or width.
• Step 4: Plot the transformed key points and sketch the graph.

Applications in Technology and Engineering

Stretching and compressing graphs are integral in designing systems and analyzing data.

• Signal Processing: Modifying signal amplitudes for transmission.
• Structural Engineering: Analyzing load distributions and stresses.
• Computer Graphics: Scaling images and models for visual representation.

Advanced Topics: Nonlinear Transformations

Beyond basic stretching and compressing, nonlinear transformations involve more complex alterations to graphs.

• Example: Exponential stretching where the rate changes dynamically.

Practice Problems

Engaging with practice problems enhances mastery of stretching and compressing concepts.

1. Given the function $f(x) = x^3$, apply a vertical stretch by a factor of 4 and describe the changes.
2. Compress the graph of $g(x) = \sqrt{x}$ horizontally by a factor of 2 and state the new function.
3. Combine both vertical compression by 0.5 and horizontal stretch by 3 for the function $h(x) = \cos(x)$ and write the transformed equation.

Solutions:

1. The transformed function is $y = 4x^3$. The graph becomes taller, increasing the rate at which $y$ grows as $x$ moves away from 0.
2. The transformed function is $y = \sqrt{\frac{x}{2}}$. The graph becomes narrower, reaching the same $y$-values more quickly as $x$ increases.
3. The transformed function is $y = 0.5\cos\left(\frac{x}{3}\right)$. The graph is vertically compressed, reducing its amplitude, and horizontally stretched, increasing its period.

Comparison Table

Summary and Key Takeaways