Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph \(f(x)=3 / x^{2}\) and the function \(g\) in the same viewing window. Describe the relationship between the two graphs. $$g(x)=-2 f(x)$$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = -2f(x)\) is a reflection of the function \(f(x) = 3 / x^{2}\) over the x-axis. While \(f(x)\) is always positive, \(g(x)\) is always negative.

Step 1: The first function

The first function to graph is \(f(x) = 3 / x^{2}\). This function represents a hyperbola with the x-axis as its asymptote. This function is never negative as the denominator, \(x^2\), is always positive and it's multiplied by 3, a positive number.

Step 2: The second function

Next, plot the function \(g(x) = -2f(x)\). Since \(f(x) =3 / x^{2}\), you need to substitute \(f(x)\) in \(g(x) = -2f(x)\) which becomes \(g(x) = -2(3 / x^{2})\). So, \(g(x)\) is -6 divided by \(x^2\) which is a reflection of \(f(x)\) since a negative sign is present.

Step 3: Relationship between the graphs

Once the graphs are drawn, it can be seen that \(g(x)\) is a reflection of \(f(x)\) over the x-axis since \(g(x) = -2f(x)\). As \(f(x)\) is always positive, \(g(x)\) is always negative because of the negative sign in \(g(x)\). The graphs of the functions are symmetric with respect to the x-axis.

Key Concepts

Hyperbola

In mathematics, a hyperbola is a type of smooth curve lying in a plane. It is an important concept when graphing rational functions like the one in this exercise. For the function \( f(x) = \frac{3}{x^2} \), this specific hyperbola does not go near the x-axis. Instead, it has the x-axis as a horizontal asymptote. A horizontal asymptote is a line that the graph of the function approaches but never touches as the input \( x \) increases or decreases without bound.
Typically, hyperbolas have a distinctive mirrored shape around what are known as their axes of symmetry. When you're plotting such a graph, it's important to understand that it will be U-shaped, opening upwards or downwards depending on the sign of the leading coefficient. Because \( f(x) \) is always positive, this hyperbola opens upwards.

Reflection

Reflection in mathematics is a type of transformation, meaning we flip a figure over a specific line, such as the x-axis. This exercise involves reflecting the graph of one function to obtain another function by modifying its equation.
Function \( g(x) = -2f(x) \) can be found by multiplying the entire \( f(x) \) by \(-2\). The negative sign results in a reflection over the x-axis. This is because multiplying the output of a function by a negative number turns all positive values into negative values, thereby flipping the graph upside down. This type of transformation is critical in analyzing and sketching graphs, as it helps understand how altering a function affects its graphical representation.

X-axis Symmetry

When discussing x-axis symmetry, we refer to a condition where the shape of the graph is mirrored along the x-axis. For the functions in this exercise, \( f(x) = \frac{3}{x^2} \) and \( g(x) = -\frac{6}{x^2} \), one is the exact x-axis mirror of the other.
Because \( g(x) \) is essentially the negative of a scaled version of \( f(x) \), it demonstrates this exact symmetry. Any point on the graph of \( f(x) \) is mirrored below the x-axis by a corresponding point on \( g(x) \). This symmetry helps understand the positional relationship between two related graphs and is particularly noticeable when using transformations like reflections across an axis.

Graphing Utility

A graphing utility is a software tool or calculator that helps visualize mathematical equations and their respective graphs. In exercises like this, graphing utilities offer a valuable means to confirm your manual graph sketches or to explore functions further.
Using a graphing utility, students can input the functions \( f(x) = \frac{3}{x^2} \) and \( g(x) = -\frac{6}{x^2} \) to see how they relate visually. You can adjust the viewing window to better capture the overall shape and behavior of the functions. This practice aids in verifying understanding and observing graphical characteristics such as asymptotes, symmetry, and reflections that might not be immediately obvious from the equations alone. It is a practical way to explore complex graph concepts in an accessible manner.