Question

Identify the appropriate base rational function, \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}},\) and then use transformations of its graph to sketch the graph of each of the following functions. Identify the asymptotes. a) \(y=\frac{1}{x+2}\) b) \(y=\frac{1}{x-3}\) c) \(y=\frac{1}{(x+1)^{2}}\) d) \(y=\frac{1}{(x-4)^{2}}\)

Short answer

The transformations are horizontal shifts, and the asymptotes are: (a) \(x = -2\), (b) \(x = 3\), (c) \(x = -1\), and (d) \(x = 4\); all have \(y=0\).

Step-by-step solution

Understand the base functions

The base functions to consider are \(y = \frac{1}{x}\) and \(y = \frac{1}{x^2}\). Their graphs and the location of their asymptotes will help in transforming other functions.

Identify the function type for (a)

For \(y=\frac{1}{x+2}\), compare it to the base function \(y = \frac{1}{x}\). The term \(x + 2\) indicates a horizontal shift of -2. This means the vertical asymptote moves from \(x=0\) to \(x=-2\). The horizontal asymptote remains at \(y=0\).

Identify the function type for (b)

For \(y=\frac{1}{x-3}\), compare it to the base function \(y = \frac{1}{x}\). The term \(x - 3\) indicates a horizontal shift of +3. This means the vertical asymptote moves from \(x=0\) to \(x=3\). The horizontal asymptote remains at \(y=0\).

Identify the function type for (c)

For \(y=\frac{1}{(x+1)^2}\), compare it to the base function \(y = \frac{1}{x^2}\). The term \(x + 1\) inside the square indicates a horizontal shift of -1. This means the vertical asymptote moves from \(x=0\) to \(x=-1\). The horizontal asymptote remains at \(y=0\).

Identify the function type for (d)

For \(y=\frac{1}{(x-4)^2}\), compare it to the base function \(y = \frac{1}{x^2}\). The term \(x - 4\) inside the square indicates a horizontal shift of +4. This means the vertical asymptote moves from \(x=0\) to \(x=4\). The horizontal asymptote remains at \(y=0\).

Sketch the graphs

Use the identified transformations to sketch the graphs of each transformed function. Reflect the horizontal shifts and identify the vertical and horizontal asymptotes on each graph. All functions will have horizontal asymptotes at \( y = 0 \). Specifically address the vertical asymptotes: \(x = -2\) for (a), \(x = 3\) for (b), \(x = -1\) for (c), and \(x = 4\) for (d).

Key concepts

horizontal shift

A horizontal shift in a rational function involves moving the graph left or right along the x-axis. When you see an equation like \(y = \frac{1}{x + k}\), the \(k\) term tells you where to shift the graph. If \(k\) is positive, move the graph to the left by \(k\) units. If \(k\) is negative, move it to the right by \(k\) units. For example, in \(y = \frac{1}{x-3}\), you shift the graph 3 units to the right because \(-3\) indicates a horizontal shift of +3 units.

vertical asymptote

Vertical asymptotes are lines the graph approaches but never touches or crosses. They occur where the denominator of a rational function equals zero. For \(y = \frac{1}{x+2}\), the vertical asymptote is at \(x = -2\) because \(x + 2 = 0\) creates a discontinuity at \(x = -2\). Note that every transformation in the form \(y = \frac{1}{x+k}\) will move the vertical asymptote to \(x = -k\). So, identifying these asymptotes is key for accurately sketching your graph.

graph transformations

Graph transformations can include shifts, stretches, and reflections. The most common transformation for rational functions are horizontal shifts, which we’ve discussed. Another transformation is vertical stretching or compressing, which happens when the function includes a coefficient affecting the numerator. For instance, examining \(y = \frac{c}{x+k}\) where \(c\) is a number. While \(c\) doesn't create a shift, it does change the steepness of the graph. In the exercises provided, we focused on horizontal shifts.

base functions

Base functions for this exercise are the simple, foundational forms of rational functions, given as \(y = \frac{1}{x}\) and \(y = \frac{1}{x^2}\). Understanding these is crucial before applying any transformations. The graph of \(y = \frac{1}{x}\) has a characteristic shape with asymptotes at \(x = 0\) (vertical) and \(y = 0\) (horizontal). For \(y = \frac{1}{x^2}\), the graph approaches similar asymptotes but with the function value always positive, creating a different shape known as a hyperbola.

asymptotes in rational functions

Asymptotes in rational functions include both vertical and horizontal asymptotes. Vertical asymptotes occur where the denominator equals zero, causing the function to approach infinity. For example, \(y = \frac{1}{x+1}\) has a vertical asymptote at \(x = -1\). Horizontal asymptotes describe the behavior of the graph as \(x\) approaches infinity or negative infinity. In rational functions like \(y = \frac{1}{x}\), the horizontal asymptote is always at \(y = 0\). Both types of asymptotes are critical for understanding the overall shape and behavior of the graph.