Problem 27 - A Graphical Approach to College Algebra

Question

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}}\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=-\frac{2}{x^{2}}$$

Step-by-Step Solution

Answer

The function is a vertical stretch, reflection of \( y = \frac{1}{x^2} \); domain: \( x \neq 0 \), range: \( (-\infty, 0) \).

Step 1: Identify the Base Function

The given function is \( f(x) = -\frac{2}{x^2} \). We need to identify the base function. Here, the base function is \( y = \frac{1}{x^2} \).

Step 2: Apply Vertical Stretch and Reflection

The transformation from the base function \( y = \frac{1}{x^2} \) to the function \( f(x) = -\frac{2}{x^2} \) involves a vertical stretch by a factor of 2, followed by a reflection across the x-axis. This is seen in the multiplication factor \(-2\).

Step 3: Graph the Transformation

To graph \( f(x) = -\frac{2}{x^2} \), take the graph of \( y = \frac{1}{x^2} \), stretch it vertically by a factor of 2, and then reflect it over the x-axis. As a result, the graph will open downwards, forming a hyperbola with its vertex at the origin.

Step 4: Find the Domain

The function \( f(x) = -\frac{2}{x^2} \) is undefined when \( x = 0 \) because division by zero is undefined. Thus, the domain is all real numbers except zero, i.e., \( x eq 0 \).

Step 5: Find the Range

Since the function is reflected and stretched, the values of \( f(x) \) are all negative numbers. The range is \( (-\infty, 0) \), indicating all negative real numbers.

Key Concepts

Base Function

In the world of transformations in mathematics, the base function serves as the starting point. It provides the simplest form of the given function, which we then modify to match our target function. For instance, consider the function \( f(x) = -\frac{2}{x^2} \).
Here, the base function is \( y = \frac{1}{x^2} \). This base function represents a fundamental shape ‒ in this case, a hyperbola ‒ that we use as the foundation of our transformations.
Understanding the base function is crucial because all transformations (like stretching, reflecting, etc.) are performed relative to this initial form. Knowing it allows you to predict how changes to the equation will affect the graph's shape.

Vertical Stretch

A vertical stretch is a transformation that magnifies the graph of a function in the vertical direction. It changes the "height" of the graph relative to the x-axis without altering its width. To achieve this, every y-coordinate gets multiplied by the stretch factor. In our example function \( f(x) = -\frac{2}{x^2} \), the number 2 indicates a vertical stretch by a factor of 2. It means that the graph of the base function \( y = \frac{1}{x^2} \) will be stretched vertically so that it becomes taller, with each output value doubled compared to the original.
This kind of transformation makes the graph steeper, enhancing the curvature's prominence, which can significantly change how the function's behavior is perceived.

Reflection

Reflections are another type of transformation where the graph of a function is flipped over a particular axis. In this example, the function \( f(x) = -\frac{2}{x^2} \) includes a negative sign, which indicates a reflection over the x-axis.
Reflections invert the graph, making all positive outputs negative and vice versa. This means for the function \( y = \frac{1}{x^2} \), every point on the graph is reflected downwards, leading to \( f(x) \) opening downward as opposed to the upward opening of the base function. As a result, the entire curve takes on negative values, altering both its direction and visual representation.

Domain and Range

Understanding the domain and range of a function is critical in identifying where the function is defined and the output values it can achieve. For \( f(x) = -\frac{2}{x^2} \), the domain consists of all real numbers except where the division becomes undefined. Here, that point is \( x = 0 \) because you cannot divide by zero. Hence, the domain is all real numbers except zero, expressed as:

• Domain: \( x eq 0 \)

The range of a function refers to all possible output values. Since the graph undergoes both a vertical stretch and a reflection, \( f(x) \) only yields negative values. These stretch infinitely towards negative infinity but never reach zero.
Therefore, the range can be articulated as:

• Range: \((-\infty, 0)\)

This understanding helps predict how the transformations affect both graphically and numerically.