Problem 47 - Algebra and Trigonometry

Question

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$ h(x)=\frac{1}{x}+2 $$

Step-by-Step Solution

Answer

The function \(h(x)= \frac{1}{x} + 2\) is a transformation of the base function \(f(x)= \frac{1}{x}\), with a shift upward by two units. This is represented graphically by lifting the graph of \(f(x)= \frac{1}{x}\) by two units. The graph is a hyperbola with asymptotes at x = 0 (vertical) and y = 2 (horizontal).

Step 1: Identify the Base Function

In the function \(h(x) = \frac{1}{x} + 2\), the base function that to we need to identify in this transformation would be \(f(x)=\frac{1}{x}\). We can observe \(f(x)\) is a rational function with the graph resembling a hyperbola with asymptotes at \(x = 0\) and \(y = 0\).

Step 2: Understand the Transformation

The function \(h(x)= \frac{1}{x} + 2\) includes a \(+2\) transformation to the base function. This transformation means that every \(y\)-value on the graph of the base function will shift upwards by two units. So, essentially, we are lifting the entire graph of the base function, \(f(x)=\frac{1}{x}\), upwards by two units.

Step 3: Graph the Transformed Function

To graph the given function \(h(x)= \frac{1}{x} + 2\), start by drawing the graph of the base function \(f(x)=\frac{1}{x}\). Then, shift the entire graph up by two units. This will provide the graph for \(h(x)= \frac{1}{x} + 2\). The new graph will still resemble a hyperbola, but the horizontal asymptote will now be at \(y = 2\) instead of \(y = 0\), due to the shift upward. The vertical asymptote remains at \(x = 0\).

Key Concepts

Function Transformations

In mathematics, function transformations allow us to manipulate a given function's graph, creating variations based on the original, or 'base,' function. For the function given, \( h(x) = \frac{1}{x} + 2 \), the transformation involves adding 2 to the base function \( f(x) = \frac{1}{x} \). This specific transformation is known as a vertical shift. A vertical shift occurs when you add or subtract a constant value from the function.

• If you add a positive number, the graph moves up.
• If you subtract a positive number, the graph moves down.

In our example, adding 2 means every point on the original graph moves 2 units higher along the y-axis. The shape of the graph remains the same; only its position changes. Understanding transformations is a crucial skill in graphing complex functions efficiently.

Asymptotes

Asymptotes are invisible lines that a graph gets infinitely close to but never actually touches or crosses. They are critical in rational functions like \( f(x) = \frac{1}{x} \) because they dictate the behavior of the graph as it approaches certain values. For the function \( h(x) = \frac{1}{x} + 2 \), there are two primary types of asymptotes:

• Vertical Asymptote: This occurs at \( x = 0 \) for both \( f(x) \) and \( h(x) \). The graph will never touch or cross this line as \( x \) approaches zero. This happens because dividing by zero creates undefined points.
• Horizontal Asymptote: For \( f(x) = \frac{1}{x} \), the horizontal asymptote is \( y = 0 \). However, the transformation in \( h(x) \) shifts it to \( y = 2 \). This pseudo-horizontal barrier illustrates how the function behaves as \( x \) grows increasingly positive or negative.

Understanding these lines helps us predict and verify the shape of a function when graphing.

Hyperbola

A hyperbola is a type of curve on a graph formed by a rational function, particularly those involving \( \frac{1}{x} \) or transformations of this form. The graph of the function \( f(x) = \frac{1}{x} \) initially resembles a hyperbola with branches in opposite quadrants of the coordinate plane. This shape emerges from the reciprocal behavior as \( x \) values grow large or approach zero.When we analyze the function \( h(x) = \frac{1}{x} + 2 \), we anticipate a similar hyperbolic structure. However, the upward shift by 2 units modifies the standard position, making the horizontal asymptote move to \( y = 2 \). The overall hyperbolic shape remains intact, maintaining its distinct open curves that decline as \( x \) moves away from zero.The hyperbola’s characteristics include:

• Two separate curves that approach both the vertical and horizontal asymptotes.
• Symmetry about the origin yet shifted along the y-axis in this transformed function.

The graphing of hyperbolas aids in visualizing the infinite nature and restrictions of a rational equation.

Base Function Identification

Identifying the base function is the foundation of understanding transformations and graphing any derived function accurately. In the problem \( h(x) = \frac{1}{x} + 2 \), the base function is \( f(x) = \frac{1}{x} \). Recognizing this base rational function informs us of its graph shape—typically hyperbolic with specific asymptotes.Steps to identify the base function:

• Look for recognizable components or simplified forms in your given function. Here, \( \frac{1}{x} \) clearly stands out.
• Understand the typical graph of this base. For \( \frac{1}{x} \), expect a hyperbola with asymptotes along the axes.
• Observe any modifications to the function, such as additional constants which indicate transformations.

With a known base, you can more easily predict and visualize transformations like shifts or stretches, essential tools in plotting rational functions accurately.