Question

Compare the graph of \(f(x)=1 / x\) with the graph of \(g\). $$g(x)=-f(x+1)=-\frac{1}{x+1}$$

Short answer

The graph of function \(g(x)\) is a mirror image of the graph of function \(f(x)\) over the x-axis and shifted 1 unit to the left.

Step-by-step solution

Sketch the original function

Firstly, the function \(f(x) = 1/x\) needs to be sketched. Note that it's hyperbola with two parts in the 1st and 3rd quadrants of the Cartesian plane. As x approaches to 0, the value of \(f(x)\) tends to infinity which contributes to a vertical asymptote at \(x = 0\). Also, as x increases or decreases without bound, \(f(x)\) tends to 0, giving a horizontal asymptote at \(y = 0\).

Identify the transformation

Now, take a look at function \(g(x)\). It is \(f(x)\) with two transformations applied. Firstly, \(x\) is replaced by \(x+1\) which represents a shift of the graph to the left by 1 unit. Secondly, the function is negated which causes a reflection in the x-axis.

Sketch the transformed function

Using the transformations identified, the function \(g(x) = -1/(x+1)\) can be sketched. The vertical asymptote shifted to x = -1 and the horizontal asymptote still at y = 0. The sections of the graph now lie in the 2nd and 4th quadrants due to reflection in the x-axis.

Compare the graphs

It can be observed from the graphs of the functions that \(g(x)\) is a transformation of \(f(x)\), shifted 1 unit to the left and reflected over the x-axis. The shapes of the graph are the same, but their positions and orientations within the coordinate plane are different.

Key concepts

Graph of a Hyperbola

A hyperbola is a type of smooth curve lying in a plane, defined by a specific equation. For the function \(f(x) = \frac{1}{x}\), it takes a hyperbolic shape.
Typically, the graph is in two parts, located in opposite quadrants. In this case, you see it situated in the 1st and 3rd quadrants.

Understanding the behavior as \(x\) approaches different key values is essential. As \(x\) moves towards zero, the value of \(f(x)\) increases to infinity. This demonstrates a vertical asymptote at \(x = 0\).
As \(x\) continues to increase or decrease towards infinity, the value of \(f(x)\) approaches zero, establishing a horizontal asymptote at \(y = 0\).

• The hyperbolic nature means the function does not touch the axes but gets infinitely close.
• Graph symmetry about the origin is a key feature.

This basic understanding of a hyperbola is crucial when moving to transformations because it helps predict how a change will manifest.

Vertical and Horizontal Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. They provide a framework for understanding the behavior of a graph at extreme values.

For the function \(f(x) = \frac{1}{x}\), the vertical asymptote is at \(x = 0\), because the function approaches infinity as \(x\) gets closer to 0 from both sides. This tells us that the graph turns back in itself as it approaches this line.
The horizontal asymptote is at \(y = 0\), as \(f(x)\) approaches zero without ever reaching it as \(x\) moves towards positive or negative infinity.

• Vertical asymptotes often arise from denominators equaling zero.
• Horizontal asymptotes illustrate end behavior—what happens to \(f(x)\) as \(x\) grows very large.

For the transformed function \(g(x) = -\frac{1}{x+1}\), we notice that:

• The vertical asymptote is now shifted to \(x = -1\) due to the \(+1\) inside the denominator.
• The horizontal asymptote remains at \(y = 0\).

Recognizing these changes helps anticipate the transformed graph’s shape beforehand.

Reflections and Shifts in Graphs

Transformations such as reflections and shifts are powerful tools for altering function graphs while maintaining their basic properties. These modifications allow you to visualize how adjustments to the equation affect the graph's placement and shape.

In the problem given, for \(g(x) = -\frac{1}{x+1}\):

• The replacement of \(x\) by \(x+1\) indicates a horizontal shift to the left by 1 unit. This impacts the graph by moving all elements one unit left, including the vertical asymptote.
• The negative sign in front reflects the graph over the x-axis. This changes the quadrants in which the graph resides—shifting it from the 1st and 3rd quadrants (original \(f(x)\) graph) to the 2nd and 4th quadrants (transformed \(g(x)\) graph).

By leveraging reflections, we can invert the graph across an axis—flipping it upside down, for example. Shifts allow positional adjustments on the coordinate plane—left, right, up, or down.
These transformations are fundamental, offering an easy method to visualize how graphical changes relate to algebraic modifications.