Stretching and compressing functions are fundamental transformations in precalculus, particularly within the study of polynomial and rational functions. These transformations alter the graph of a function, affecting its shape and position without changing its basic structure. Understanding these concepts is crucial for Collegeboard AP students as they form the foundation for more complex function analysis and graphing techniques.

Key Concepts

Understanding Function Transformations

Function transformations involve shifting, reflecting, stretching, and compressing the graph of a function. These operations modify the function's graph in a systematic way, allowing for a deeper understanding of its behavior and properties.

Stretching Functions Vertically

Vertical stretching occurs when a function is multiplied by a constant greater than one, which makes the graph taller. Mathematically, if we have a function f(x), stretching it vertically by a factor of a (>1) results in the function af(x).

Example: Consider the function f(x) = x². Stretching it vertically by a factor of 2 gives 2x². The graph becomes narrower as its y-values are doubled.

The general form for vertical stretching:

$$ g(x) = a \cdot f(x) $$ where a > 1.

Compressing Functions Vertically

Vertical compression occurs when a function is multiplied by a constant between 0 and 1, making the graph shorter. For a function f(x), compressing it vertically by a factor of a (0 < a < 1) results in the function af(x).

Example: Taking f(x) = x² and compressing it vertically by a factor of 0.5 gives 0.5x². The graph becomes wider as its y-values are halved.

The general form for vertical compression:

$$ g(x) = a \cdot f(x) $$ where 0 < a < 1.

Stretching Functions Horizontally

Horizontal stretching occurs when the input of a function is multiplied by a factor between 0 and 1, effectively stretching the graph horizontally. For the function f(x), stretching it horizontally by a factor of b (0 < b < 1) results in f(bx).

Example: If f(x) = \sqrt{x}, stretching it horizontally by a factor of 0.5 gives f(0.5x) = \sqrt{0.5x}. The graph spreads out along the x-axis.

The general form for horizontal stretching:

$$ g(x) = f(bx) $$ where 0 < b < 1.

Compressing Functions Horizontally

Horizontal compression occurs when the input of a function is multiplied by a factor greater than one, making the graph narrower. For f(x), compressing it horizontally by a factor of b (>1) results in f(bx).

Example: For f(x) = \sqrt{x}, compressing it horizontally by a factor of 2 gives f(2x) = \sqrt{2x}. The graph becomes steeper along the x-axis.

The general form for horizontal compression:

$$ g(x) = f(bx) $$ where b > 1.

Combined Transformations

Often, stretching and compressing transformations are combined with other transformations such as translations (shifts) and reflections. The order of transformations can affect the final graph, so it's essential to apply them systematically.

Example: Consider the function f(x) = x³. Applying a vertical stretch by a factor of 3 and a horizontal compression by a factor of 2 results in g(x) = 3f(2x) = 3(2x)³ = 24x³.

Impacts on Function Properties

Stretching and compressing functions impact various properties such as amplitude, period, and rate of growth or decay. For example, vertically stretching a sine function increases its amplitude, while horizontally compressing it decreases its period.

Example: The sine function f(x) = \sin(x) has an amplitude of 1. Stretching it vertically by 2 yields 2\sin(x), increasing the amplitude to 2.

Graphing Stretching and Compressing Transformations

When graphing transformed functions, it's helpful to identify key points and apply the transformations systematically. Start by plotting the basic function, then apply vertical and horizontal stretches or compressions, followed by any translations or reflections.

Example: To graph g(x) = 2(x - 1)², follow these steps:

Start with the basic function f(x) = x².
Translate the graph 1 unit to the right to get f(x) = (x - 1)².
Apply a vertical stretch by a factor of 2 to obtain g(x) = 2(x - 1)².

The resulting graph is narrower and shifted to the right compared to the basic parabola.

Applications of Stretching and Compressing Functions

Understanding stretching and compressing is essential for modeling real-world scenarios where relationships between variables change in scale. For instance, in physics, these transformations can model changes in velocity or acceleration, while in economics, they can represent scaling of cost functions.

Example: If a company's profit function is P(x) = \sqrt{x}, applying a vertical stretch by 3 results in P(x) = 3\sqrt{x}, indicating increased profitability at each production level.

Common Mistakes to Avoid

When performing stretching and compressing transformations, students often confuse the direction of stretching and compression, especially with horizontal transformations. Remember that multiplying the input by a factor affects the horizontal scale inversely.

Tip: For horizontal transformations, use the rule: a factor less than 1 stretches the graph, while a factor greater than 1 compresses it.

Practice Problems

To reinforce understanding, consider the following practice problems:

Given f(x) = \sin(x), graph g(x) = 2\sin(\frac{1}{2}x).
If h(x) = \ln(x), apply a horizontal compression by a factor of 3 and a vertical stretch by a factor of 0.5. Write the transformed function.
Describe the transformations applied to f(x) = \frac{1}{x} to obtain g(x) = -2f(4x).

Solutions:

Vertical stretch by 2 and horizontal stretch by 2.
g(x) = 0.5\ln(4x).
Vertical stretch by 2, horizontal compression by 4, and reflection over the x-axis.

Comparison Table

Aspect	Vertical Stretch	Vertical Compression
Definition	Multiplying the function by a constant > 1, making the graph taller.	Multiplying the function by a constant between 0 and 1, making the graph shorter.
Effect on Equation	$g(x) = a \cdot f(x)$ where $a > 1$	$g(x) = a \cdot f(x)$ where $0 < a < 1$
Graphical Impact	Narrower graph, taller peaks and deeper troughs.	Wider graph, shorter peaks and shallower troughs.
Example	From $f(x) = x²$ to $g(x) = 2x²$.	From $f(x) = x²$ to $g(x) = 0.5x²$.
Applications	Modeling increased rates, such as acceleration.	Modeling decreased rates, such as slowing growth.

Summary and Key Takeaways

Stretching and compressing functions modify the graph's height and width without altering its fundamental shape.
Vertical transformations affect the y-values, while horizontal transformations impact the x-values.
Combining transformations requires careful application to maintain accuracy in graphing.
Understanding these concepts is essential for analyzing and modeling real-world functions in precalculus.

Examiner Tip

Tips

To master function transformations for the AP exam, always start by identifying the basic function. Use mnemonic devices like "V for Vertical, H for Horizontal" to remember which factor affects which axis. Practice sketching graphs step-by-step to visualize each transformation clearly. Additionally, double-check your work by plugging in specific x-values to ensure transformations are correctly applied.

Did You Know

Stretching and compressing functions aren't just mathematical concepts—they're essential in various fields. For example, in engineering, these transformations help model stress and strain in materials. Additionally, in computer graphics, they are used to scale images and animations seamlessly. Surprisingly, the principles of function transformations are also applied in music, where altering waveforms affects sound pitch and volume.

Common Mistakes

Students often confuse vertical and horizontal transformations. For instance, multiplying by a factor greater than 1 vertically stretches a graph, but the same factor horizontally compresses it. Another common error is neglecting to apply negative signs correctly, leading to incorrect reflections. Additionally, forgetting the order of operations when combining multiple transformations can result in inaccurate graphs.

FAQ

What is the difference between vertical and horizontal stretching?

Vertical stretching affects the y-values by multiplying the entire function by a constant, making the graph taller or shorter. Horizontal stretching involves multiplying the input variable by a constant, which stretches or compresses the graph along the x-axis.

How do you determine the factor for stretching or compressing?

The factor is determined by the constant multiplied to the function (vertical) or the input variable (horizontal). A factor greater than 1 indicates a stretch, while a factor between 0 and 1 indicates a compression.

Can stretching and compressing affect the domain and range of a function?

Stretching and compressing vertically do not change the domain but can alter the range. Horizontal transformations can affect both the domain and range depending on the function and the type of transformation applied.

How do reflections work with stretching and compressing?

Reflections invert the graph over the specified axis. For vertical reflections, multiply the function by -1, and for horizontal reflections, multiply the input by -1. These can be combined with stretching and compressing by applying the respective constants.

Why is the order of transformations important?

The order of transformations affects the final graph because each transformation builds upon the previous one. Changing the order can lead to different results, so it's crucial to apply them systematically, usually starting with horizontal transformations before vertical ones.

How can I check if my transformed graph is accurate?

You can verify accuracy by selecting specific x-values, applying the transformations to these values, and ensuring the corresponding y-values match the transformed function. Additionally, comparing key features like intercepts, vertices, and asymptotes can help confirm correctness.

1. Functions Involving Parameters, Vectors and Matrices

1.1 Parametric Functions

1.1.1 Graphing parametric functions in x-y planes

1.1.2 Comparing parametric and Cartesian forms

1.1.3 Identifying parametric equations for simple curves

1.2 Parametric Functions and Rates of Change

1.2.1 Calculating derivatives of parametric equations

1.2.2 Understanding tangent lines in parametric graphs

1.2.3 Exploring relationships between x and y changes

1.3 Vectors

1.3.1 Adding and subtracting vectors

1.3.2 Computing magnitudes and directions