Problem 29 - College Algebra and Calculus: An Applied Approach

Question

Compare the graph of \(f(x)=8 / x^{3}\) with the graph of \(g\). $$g(x)=-f(x)=-\frac{8}{x^{3}}$$

Step-by-Step Solution

Answer

The graph of \(g(x)=-\frac{8}{x^3}\) is a reflection of the graph of \(f(x)=\frac{8}{x^3}\) across the x-axis.

Step 1: Graph of \(f(x)\)

To start, the function \(f(x)=\frac{8}{x^3}\) needs to be graphed. This function is an example of a hyperbola because it is expressed as a reciprocal of a power function. The behavior of the graph of this function can be summarized as follows: as \(x\) approaches 0 from the positive side, \(f(x)\) approaches positive infinity and as \(x\) approaches 0 from the negative side, \(f(x)\) approaches negative infinity. As \(x\) approaches positive or negative infinity, \(f(x)\) becomes closer to 0.

Step 2: Transformation of \(f(x)\)

Given that \(g(x) = -f(x)\), it means that every point on the graph of \(f(x)\) will be multiplied by -1. This flips the graph of \(f(x)\) around the x-axis. Parts of the graph of \(f(x)\) in the first and second quadrants, where \(y\) is positive, will be moved to the third and fourth quadrants, where \(y\) is negative. And parts of \(f(x)\) in the third and fourth quadrants, where \(y\) is negative, will be moved to the first and second quadrants, where \(y\) is positive.

Step 3: Graph of \(g(x)\)

Now, the graph of the function \(g(x)=-\frac{8}{x^3}\) can be plotted. As \(x\) approaches 0 from the positive side where \(f(x)\) was approaching positive infinity, \(g(x)\) instead approaches negative infinity due to the multiplication by -1. Similarly, as \(x\) approaches 0 from the negative side, \(g(x)\) approaches positive infinity.

Step 4: Comparison of the graphs

Finally, compare the two graphs. The graph of \(g(x)\) is a reflection of the graph of \(f(x)\) across the x-axis. This is because multiplying by -1 flips the graph over the x-axis.

Key Concepts

Graph Reflection

When we talk about graph reflection, it's all about flipping a graph over a certain axis, much like turning a mirror image upside down. In our exercise, the graph of the function from \(f(x)\) to \(g(x)\) involves reflection. Here, the graph of \(f(x)\) is flipped over the x-axis because of the negative sign in \(g(x)\). This transformation results in each y-value on the graph of \(f(x)\) being multiplied by -1, flipping the entire graph downward.

• Positive parts of \(f(x)\) become negative in \(g(x)\).
• Negative parts of \(f(x)\) become positive in \(g(x)\).

Graph reflection is a simple transformation that reveals how the graph changes position but retains its shape. It's like turning the whole picture upside down around the x-axis, neatly switching everything below the axis to above, and vice versa. This technique helps in understanding symmetry and transformations in functions.

Hyperbola

A hyperbola is a type of mathematical curve characterized by two separate, mirror-like branches. In the context of the function \(f(x) = \frac{8}{x^3}\), it can be seen as part of a family known as reciprocal functions. Hyperbolas have a distinct feature: as the variable \(x\) approaches a specific value, the output of the function tends to infinity.

• As \(x\) gets closer to 0, \(f(x)\) tends to either positive or negative infinity.
• As \(x\) moves towards infinity or negative infinity, \(f(x)\) approaches 0.

The shape created is an open curve with two branches curving away from each other. Hyperbolas often represent division by zero scenarios, leading their graphed lines to shoot upwards or downwards, indicating dramatic changes in value.

Reciprocal Function

A reciprocal function is a special type of function where the output value of a number is flipped over its input value, mathematically represented as \(\frac{1}{x^n}\). In our given function \(f(x) = \frac{8}{x^3}\), it's a reciprocal of a power function. The key aspect of reciprocal functions is their asymptotic nature, meaning they tend to approach a line asymptotically without actually touching it.

• Reciprocal functions often have vertical and horizontal asymptotes.
• They tend towards zero as \(x\) grows larger, despite never reaching it.

This peculiar behavior shows how reciprocal functions spread wide apart near zero and grow closely parallel to the x-axis far from zero. This tendency creates the hyperbolic appearance in their graph, with dramatic declines and rises based on their reciprocal nature.

Power Function

Power functions are basic functions of the form \(y = a x^n\), where the value of n determines the degree of the function. The given function, \(f(x) = \frac{8}{x^3}\), can also be interpreted through its power component. The denominator \(x^3\) contributes this power aspect.

• The power \(n=3\) affects the growth and decay rate of the function.
• The power element makes the graph steeper for higher values of \(n\).

Power functions play a crucial role in shaping the reciprocal function form. Each change in the power value results in a different kind of graph behavior. They determine how quickly the graph reaches its limit or how steeply it climbs or falls, especially around the asymptotic regions.