Problem 32 - 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. $$f(x)=\frac{-1}{(x-4)^{2}}+2$$

Step-by-Step Solution

Answer

The graph is a vertically reflected hyperbola shifted right to \(x = 4\) and up to \(y = 2\). Domain: \(x \neq 4\); Range: \(y < 2\).

Step 1: Identify Basic Graph

The starting point is the function \(y = \frac{1}{x^2}\), which is the graph of a hyperbola that approaches the x-axis and y-axis but never touches them. It is symmetric about the y-axis and lies in the first and second quadrants.

Step 2: Transformation - Horizontal Shift

The expression \((x - 4)\) in the denominator indicates a horizontal shift. The graph of \(y = \frac{1}{x^2}\) is shifted 4 units to the right. As a result, the center of the hyperbola, originally at the origin, is now at \(x = 4\).

Step 3: Transformation - Reflection and Vertical Shift

The negative sign in \(-\frac{1}{(x-4)^2}\) indicates a reflection across the x-axis. The graph is flipped vertically. The addition of \(+2\) shifts the entire graph 2 units up. Thus, the horizontal asymptote moves from \(y = 0\) to \(y = 2\).

Step 4: Sketching the Graph

Start with the basic shape of \(y = \frac{1}{x^2}\), shift it 4 units to the right, reflect it across the x-axis, and then shift it 2 units up. Draw asymptotes at \(x = 4\) and \(y = 2\). Sketch the hyperbola accordingly.

Step 5: Determine Domain and Range

Since \(x=4\) results in division by zero, it is not included in the domain. Thus, the domain is all real numbers except \(x=4\): \(x \in \mathbb{R}, x eq 4\). The range includes all real numbers less than 2 because the graph has been reflected downward and shifted up: \(y < 2\).

Key Concepts

Graphing Techniques

Function transformations are powerful techniques which help us understand how functions change and how their graphs move. For a function like \(f(x)=\frac{-1}{(x-4)^{2}}+2\), graphing begins by identifying the base function. Here, it is \(y=\frac{1}{x^2}\), known for its hyperbolic shape.
The initial graph of \(y=\frac{1}{x^2}\) has the following characteristics:

• It is symmetric about the y-axis.
• It lies entirely above the x-axis because all values are positive.
• It has asymptotes at the x-axis \(y=0\) and the y-axis \(x=0\).

To transform this base graph into \(f(x)\), we apply several transformations:

• Horizontal Shift: The term \((x-4)\) shifts the graph 4 units to the right.
• Reflection: The negative sign causes a reflection over the x-axis.
• Vertical Shift: Adding \(+2\) moves the graph up by 2 units, changing the horizontal asymptote to \(y=2\).

Visualizing each step helps build an understanding of how graphs respond to algebraic transformations.

Domain and Range

Understanding the domain and range of a function provides insight into the limitations and possible outputs of the function. For the function \(f(x)=\frac{-1}{(x-4)^{2}}+2\), determining these aspects is crucial:

• Domain: The domain of a function is the set of all possible inputs \(x\) that the function can accept. Here, because division by zero is undefined, the point \(x=4\) must be excluded. Thus, the domain is all real numbers except \(x=4\): \(x \in \mathbb{R}, x eq 4\).
• Range: The range represents all the possible output values. Due to the reflection and vertical shift of the graph, \(f(x)\) can take any real value less than 2. Therefore, the range is \(y < 2\).

Understanding these attributes helps in analyzing the behavior of the function and predicting its outputs.

Hyperbola

A hyperbola is a fascinating type of curve that appears in various areas of mathematics and science. In the case of \(f(x) = \frac{-1}{(x-4)^2} + 2\), the hyperbolic nature stems from the term \(\frac{1}{x^2}\).
Here's what characterizes a hyperbola in this context:

• A hyperbola typically approaches its asymptotes (the x-axis and y-axis for the base \(\frac{1}{x^2}\)) but never actually reaches them.
• The transformation shifts and reflects the hyperbola to create a new set of asymptotes: one vertical at \(x=4\) and one horizontal at \(y=2\).
• In this particular function, the hyperbola opens downwards, due to the negative sign in \(-\frac{1}{(x-4)^2}\), which reflects the graph over the x-axis.

These properties make hyperbolas versatile and intriguing, with their behavior being distinctly different from other types of curves, such as circles or parabolas.