Problem 47 - College Algebra with Modeling and Visualization

Question

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=\frac{1}{x}+2 $$

Step-by-Step Solution

Answer

Graph \( g(x) = \frac{1}{x} + 2 \) with vertical asymptote at \( x=0 \) and horizontal asymptote at \( y=2 \).

Step 1: Identify the Base Function

The given function is \( g(x) = \frac{1}{x} + 2 \). The base function here is \( f(x) = \frac{1}{x} \). We can see that \( g(x) \) is derived from \( f(x) = \frac{1}{x} \) by adding 2. Thus, \( g(x) \) can be expressed as a transformation of \( f(x) \).

Step 2: Determine the Transformation

The transformation involved in \( g(x) = \frac{1}{x} + 2 \) is a vertical shift. By adding 2 to \( f(x) \), the graph of \( g(x) \) is shifted upwards by 2 units compared to \( f(x) \).

Step 3: Sketch the Graph

To sketch \( g(x) \), begin with the graph of \( f(x) = \frac{1}{x} \), which is a hyperbola with vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \), respectively. Shift this graph vertically by 2 units. The new horizontal asymptote becomes \( y = 2 \), while the vertical asymptote remains at \( x = 0 \).

Step 4: Identify Asymptotes

For the graph of \( g(x) \), there is a vertical asymptote at \( x = 0 \) because the function is undefined at \( x = 0 \). There is also a horizontal asymptote at \( y = 2 \) due to the vertical shift of the entire graph upwards by 2 units.

Key Concepts

Vertical Asymptotes

In graph transformations, a vertical asymptote is an invisible line that the graph approaches but never actually reaches or crosses. These occur when the denominator of a rational function approaches zero, making the function undefined. For the function \( f(x) = \frac{1}{x} \), the vertical asymptote is at \( x = 0 \) because division by zero is not possible. This means \( f(x) \) becomes undefined whenever \( x \) equals zero, resulting in an asymptotic behavior as \( x \) approaches this point from either side.

In our case with \( g(x) = \frac{1}{x} + 2 \), the transformation of the function does not impact the position of the vertical asymptote. Even though there is a vertical shift, vertical asymptotes remain at \( x = 0 \) because the denominator is unchanged. Thus, the value \( x = 0 \) creates a barrier that the curve of the function will never touch or cross, continually arcing towards it but never actually reaching. This concept is fundamental in analyzing how graphs behave around undefined values, allowing us to predict the nature of movement and limits of complex functions.

Horizontal Asymptotes

Horizontal asymptotes represent the behavior of a graph as \( x \) approaches positive or negative infinity. They indicate the value that the function will get closer to, but not necessarily reach, as the input becomes very large or very small. For the base function \( f(x) = \frac{1}{x} \), the horizontal asymptote occurs at \( y = 0 \). As \( x \) trends towards infinity or negative infinity, \( f(x) \) approaches zero.

When we modify \( f(x) \) to become \( g(x) = \frac{1}{x} + 2 \), the horizontal asymptote shifts. This is because the constant term causes the entire graph to move upwards, effectively raising the level towards which the function approaches as \( x \) becomes large or small. The horizontal asymptote of \( g(x) \) is now at \( y = 2 \). Regardless of how large \( x \) becomes, \( g(x) \) will approach 2. This shift showcases how added constants affect the end behavior of rational functions, marking the line towards which the graph gravitates indefinitely without reaching.

Vertical Shift

The concept of vertical shift in graph transformations refers to moving the entire graph of a function up or down on the coordinate plane. This is achieved by adding or subtracting a constant value to the function. For instance, when we transform \( f(x) = \frac{1}{x} \) to \( g(x) = \frac{1}{x} + 2 \), each point on the graph is elevated by 2 units, signifying a vertical shift.

A vertical shift affects the horizontal asymptote but leaves the vertical asymptote unchanged. Thus, the vertical shift in \( g(x) \) adjusts the horizontal asymptote from \( y = 0 \) to \( y = 2 \). This shifts the graph's overall position without altering the undefined points at \( x = 0 \). Vertical shifts are crucial in understanding how additions or subtractions of constants to functions impact the position of graphs. They allow us to tailor functions to better fit data or desired positions by simply adjusting the vertical location of their curves.