Question

Match the function with the correct transformation of the graph of \(f\). Explain your reasoning. \(y=2 f(x)\)

Short answer

The given function represents a vertical stretch by a factor of 2 of the graph of the base function \(f(x)\).

Step-by-step solution

Identify Transformation

The equation \(y=2 f(x)\) involves a scalar multiplication of the base function \(f(x)\) in the range. Scalar multiplication changes the shape of the graph but doesn't shift it or reflect it across any axis. In this case, the function \(f(x)\) is multiplied by a positive value bigger than one (2), that is a stretch on the vertical axis by the factor of 2.

Key concepts

Scalar Multiplication

In mathematics, scalar multiplication refers to the process of multiplying a function by a numerical constant, also known as a scalar. This operation influences the function's outputs but leaves the input unchanged. When you perform scalar multiplication on a function, like multiplying a function's output by 2, it affects the range (the y-values) directly.
The function's overall structure or shape will change, but the transformation does not result in any shifts or reflections along the graph.

• Example: Consider the function transformation represented by the equation \( y = c \, f(x) \), where \( c \) is a scalar.
• In our case, \( y = 2 f(x) \) multiplies the output of function \( f(x) \) by 2.
• The scalar \( c = 2 \) results in each y-value being doubled, thus affecting the height of the function graphically.

Understanding scalar multiplication is fundamental to dealing with more complex function transformations.

Vertical Stretch

A vertical stretch is a specific type of graph transformation where every point on the graph is pulled away from the x-axis by a factor greater than one. In simpler terms, the graph of the function becomes 'taller'.
When a function, such as \( f(x) \), is multiplied by a factor \( a \) such that \( a > 1 \), the graph undergoes a vertical stretch.

• The factor by which the function is stretched vertically is equal to \( a \).
• For \( y = 2 f(x) \), the graph of \( f(x) \) is stretched vertically by a factor of 2.
• Visually, this means that points originally at y-values on the graph are two times higher than they were before calculating \( y = f(x) \).

A vertical stretch emphasizes the variation in the function's output, making peaks and valleys more pronounced.

Function Transformations

Function transformations encompass various operations that alter the appearance of the function’s graph. These transformations include, but are not limited to, translations, stretches, compressions, and reflections. Each transformation applies differently to a function, modifying its position or shape on the coordinate plane.

• Translation: Moves the graph horizontally or vertically without changing its shape.
• Stretch/Compression: Alters the graph's width or height, like the vertical stretch we discussed.
• Reflection: Flips the graph across a given axis.

In this context, \( y = 2 f(x) \) does not translate, compress, or reflect the graph, but rather stretches it vertically by doubling all y-values.
Understanding these transformations enables one to predict and describe the changes to any given function accurately. By mastering these concepts, students can easily understand complicated graphs and their transformations.