Consider the function [tex]$h(x)=\frac{1}{x}$[/tex]. Which graph shows [tex]$h(2 x)+4$[/tex]?
Answer (1)
The function h ( x ) = x 1 is horizontally compressed to h ( 2 x ) = 2 x 1 .
The function is then vertically shifted upwards by 4 units to h ( 2 x ) + 4 = 2 x 1 + 4 .
The transformed function has a vertical asymptote at x = 0 and a horizontal asymptote at y = 4 .
The graph approaches 4 from above as x goes to ± ∞ , approaches ∞ as x approaches 0 from the right, and approaches − ∞ as x approaches 0 from the left.
Explanation
Understanding the Transformations We are given the function h ( x ) = x 1 and asked to find the graph of h ( 2 x ) + 4 . This involves two transformations: a horizontal compression and a vertical shift.
Horizontal Compression First, we find the expression for h ( 2 x ) . Since h ( x ) = x 1 , we substitute 2 x for x to get h ( 2 x ) = 2 x 1 . This represents a horizontal compression by a factor of 2.
Vertical Shift Next, we find the expression for h ( 2 x ) + 4 . We add 4 to the expression for h ( 2 x ) to get h ( 2 x ) + 4 = 2 x 1 + 4 . This represents a vertical shift upwards by 4 units.
Analyzing the Transformed Function Now, let's analyze the transformed function y = 2 x 1 + 4 . We need to determine the asymptotes and the behavior of the function as x approaches these asymptotes.
Finding the Asymptotes The vertical asymptote occurs when the denominator is zero, which is at x = 0 . The horizontal asymptote occurs as x approaches infinity. In this case, as x approaches infinity, 2 x 1 approaches 0, so y approaches 4. Thus, the horizontal asymptote is at y = 4 .
Analyzing the Function's Behavior As x approaches infinity, y approaches 4 from above (since 2 x 1 is positive for positive x ). As x approaches negative infinity, y approaches 4 from above (since 2 x 1 is negative for negative x , but the +4 shifts it above the x-axis). As x approaches 0 from the right, y approaches infinity. As x approaches 0 from the left, y approaches negative infinity.
Conclusion Therefore, the graph of h ( 2 x ) + 4 = 2 x 1 + 4 has a vertical asymptote at x = 0 and a horizontal asymptote at y = 4 . The function approaches 4 from above as x goes to ± ∞ , approaches ∞ as x approaches 0 from the right, and approaches − ∞ as x approaches 0 from the left.
Examples
Understanding transformations of functions is crucial in many fields. For example, in signal processing, compressing a signal in time corresponds to scaling the independent variable, similar to the horizontal compression in our problem. Adding a constant corresponds to shifting the signal's amplitude, similar to the vertical shift. By analyzing these transformations, engineers can manipulate signals to extract useful information or improve their quality. This concept is also applicable in image processing, where transformations are used to resize, shift, or enhance images.
