Question

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{-12}{x} $$

Short answer

The graph of function \(g(x)=\frac{-12}{x}\) is a vertically-stretched and reflected version of the graph of function \(f(x)=\frac{1}{x}\). They both share the same asymptotes at x=0 and y=0. The only difference is that g(x) is reflected along the x-axis and its graph is stretched vertically by a factor of 12.

Step-by-step solution

Understand the Original Function

Start by understanding the function \(f(x)=\frac{1}{x}\). This function, also known as a reciprocal function, is undefined at x=0 i.e., it has a vertical asymptote at x=0, and the graph of the function approaches this asymptote but never touches or crosses it. The same way it has a horizontal asymptote at y=0, and its graph approaches this asymptote as well. It generates a hyperbola with two branches located in 1st (where x > 0) and 3rd (where x < 0) quadrants. The graph is symmetric with respect to the origin.

Graph the Function \(g(x)=\frac{-12}{x}\)

Next, graph the function \(g(x)=\frac{-12}{x}\). As this function is a variant of the function \(f(x)\), it shares a similar structure, but it differs in some key ways. It will also have vertical asymptote at x=0 and horizontal asymptote at y=0. But it will generate a hyperbola with two branches located in 2nd (where x > 0) and 4th (where x < 0) quadrants because of the negative sign. The shape of the hyperbola remains the same, but the function is stretched vertically due to multiplication by 12.

Compare the Two Graphs

When comparing the two graphs, we can see that they both share the same vertical and horizontal asymptotes(x=0 and y=0). The only difference is due to the -12 in function \(g(x)\), which reflects the hyperbola in the x-axis (due to the negative sign) and stretches it vertically (due to multiplication by 12). So the branches of the graph of function \(g(x)\) are now located in 2nd (where x > 0) and 4th (where x < 0) quadrants.

Key concepts

Reciprocal Function

A reciprocal function typically has the form \( f(x) = \frac{1}{x} \), which features some interesting characteristics. Most notably, the function is not defined at \( x=0 \) because dividing by zero is undefined in mathematics. As a result, the graph of a reciprocal function has a vertical asymptote at \( x=0 \), which the graph approaches but never crosses.

This type of function also behaves interestingly as \( x \) approaches large positive or negative values; the function's value approaches zero, creating a horizontal asymptote at \( y=0 \). The graph forms a hyperbola with two distinct branches: one in the first quadrant for positive \( x \) values, and one in the third quadrant for negative \( x \) values. These branches mirror each other across the origin due to the function's odd symmetry.

Asymptote

An asymptote is a line that a function approaches but never actually reaches. Asymptotes can be horizontal, vertical, or even oblique (slanted). They give us a visual representation of certain behaviors of functions as \( x \) or \( y \) values become very large or near certain critical points. For reciprocal functions, there are always two asymptotes to consider:

• A vertical asymptote, typically where the function is undefined (in reciprocal functions, this is at \( x=0 \)).
• A horizontal asymptote, which shows the behavior as \( x \) becomes large in magnitude in either the positive or negative direction (in reciprocal functions, this occurs as \( y \) approaches zero).

These asymptotes are crucial for graphing because they frame the hyperbola and help us determine the end behavior of the function.

Hyperbola

A hyperbola is a type of curve on a graph, characteristic of reciprocal functions, defined by its two asymptote-crossed branches. When graphing a reciprocal function like \( f(x)=\frac{1}{x} \), we observe a hyperbola in the first and third quadrants. The hyperbola has two branches, each getting closer to the function's asymptotes but never touching or intersecting them.

The hyperbola for the reciprocal function is symmetric about the origin, which reflects the property that \( f(-x) = -f(x) \) for all \( x \). The curves are steeper closer to the \( x \) and \( y \) axes and flatten out as they move away from the axes, conforming to the asymptotic behavior of the function.

Function Transformation

The concept of function transformation involves altering the basic graph of a function in specific ways to obtain a new function. Common transformations include shifts, stretches, compressions, and reflections. Comparing the functions \( f(x)=\frac{1}{x} \) and \( g(x)=\frac{-12}{x} \) offers a clear example of this.

The negative sign in \( g(x) \) reflects the original graph across the \( x \) or \( y \) axis—in this case, across the \( x \) axis, flipping the hyperbola's branches from quadrants I and III to quadrants II and IV. The multiplication by 12 represents a vertical stretch, increasing the steepness of the hyperbola. This means for every \( x \) value; the \( y \) value is twelve times further away from the \( x \) axis than it is on the original function. This transformation doesn't change the location of asymptotes but alters the shape and position of the graph itself.