Problem 2 - College Algebra with Corequisite Support

Question

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?

Step-by-Step Solution

Answer

Vertical stretches multiply the function output; horizontal stretches multiply the input.

Step 1: Understanding Transformations

To identify transformation types like horizontal or vertical stretches, we need to consider the general form of function transformations: - A vertical stretch involves multiplying the entire function by a factor. If the function is \( f(x) \), and the transformation results in \( af(x) \), where \( |a| > 1 \), then it is a vertical stretch.- A horizontal stretch involves multiplying the input \( x \) by a factor. If the function is \( f(x) \), and the transformation results in \( f(bx) \), where \( 0 < |b| < 1 \) (for stretching), then it is a horizontal stretch.

Step 2: Analyzing Example of Vertical Stretch

Consider the function \( f(x) = x^2 \) and the transformation \( g(x) = 3x^2 \). Here, the entire function is multiplied by 3. Since \( |3| > 1 \), it represents a vertical stretch of the graph of \( f(x) \).

Step 3: Analyzing Example of Horizontal Stretch

Consider the function \( f(x) = x^2 \) and the transformation \( g(x) = (0.5x)^2 \). Here, \( x \) is multiplied by 0.5 inside the function. Since \( 0 < |0.5| < 1 \), it represents a horizontal stretch of the graph of \( f(x) \).

Key Concepts

Horizontal Stretch

In function transformations, a horizontal stretch can sometimes be a less intuitive concept to grasp. To perform a horizontal stretch, you multiply the argument (input) of the function by a factor that is less than 1.

For instance, given a function like \( f(x) = x^2 \), applying a transformation \( f(bx) \) where \( 0 < |b| < 1 \) will stretch the graph horizontally. What this means is, instead of the parabola getting narrower, it actually opens wider. This is because the function takes more time (or x-values) to reach the same output, effectively widening the graph.

• The numerical value of \( b \) determines the degree of stretching.
• If \( b = 0.5 \), the graph is stretched to twice its width in the direction of the x-axis.
• The original function's behavior widens accordingly based on this factor.

Understanding horizontal stretches requires practice in seeing how inputs are modified and how these modifications impact the graph.

Vertical Stretch

Vertical stretches are a bit more straightforward than horizontal stretches. A vertical stretch is achieved by multiplying the entire function by a factor greater than 1. Consider the example where the original function is \( f(x) = x^2 \). If we transform it to \( af(x) = 3x^2 \), this is a vertical stretch. Here, every output of the function is increased threefold.

The impact of this transformation:

• Causes the graph of the function to stretch upward.
• The value of \( a \) indicates how much taller or elongated the graph will appear.
• A factor \( a > 1 \) means the function’s peaks and valleys are more pronounced.

Visualizing vertical stretches helps understand how the entire function shoots up, making the graph appear taller and leaner in the vertical direction, especially at points distant from the x-axis.

Graph Transformations

Graph transformations encompass a range of alterations, including horizontal and vertical stretches, and are essential for adjusting the graph relative to its axes. They help us modify the shape and position of a graph to better fit data, analyze equations, or solve problems.

These transformations can be categorized as follows:

• Horizontal Stretch/Compress: Adjusts the width of the graph due to changes in the input variable.
• Vertical Stretch/Compress: Modifies the height of the graph as a result of scaling the function itself.
• Reflections: Flips the graph over a specified axis, altering its orientation.
• Translations: Moves the graph up, down, left, or right without changing its shape.

Combining these transformations can help graph complex functions step-by-step. Mastering graph transformations provides a robust foundation for more advanced mathematical concepts and real-world applications.