In Exercises 1-16, use the graph of y = f(x) to graph each function g.
g(x) = -f(2x) - 1

Key Concepts

Function Transformation

Function transformations involve altering the graph of a function through shifts, stretches, compressions, or reflections. In this case, the function g(x) = -f(2x) - 1 represents a vertical reflection, horizontal compression, and downward shift of the original function f(x). Understanding these transformations is crucial for accurately graphing the new function.

Horizontal Compression

A horizontal compression occurs when the input of a function is multiplied by a factor greater than 1, which in this case is the '2' in g(x) = -f(2x). This transformation effectively reduces the width of the graph, making it appear 'narrower' as the x-values are scaled down, resulting in a faster increase or decrease in the function's output.

Vertical Reflection and Shift

A vertical reflection flips the graph of a function over the x-axis, which is represented by the negative sign in g(x) = -f(2x). Additionally, the '-1' at the end of the function indicates a vertical shift downward by one unit. Together, these transformations change the orientation and position of the graph, which is essential for accurately plotting g(x).