Problem 48 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph of the given function \(h(x) = \frac{1}{x} + 1\) is a transformation of the function \(f(x) = \frac{1}{x}\), where the original graph has been shifted 1 unit upwards.

Step 1: Identify the transformations

The given equation \(h(x) = \frac{1}{x} + 1\) represents the graph of the function \(f(x) = \frac{1}{x}\) translated one unit upward. This is because the '+1' in \(h(x)\) translates the graph of \(f(x)\) one unit upward in the y-direction. There is no change in x-direction.

Step 2: Plot f(x)

To start with, plot the base function \(f(x) = \frac{1}{x}\). It is a curve that approaches but never touches the x-axis and y-axis (i.e., x=0 and y=0 are asymptotes). The curve is in the first quadrant (for x>0) and the third quadrant (for x<0).

Step 3: Apply transformation

Shift the graph of f(x) one unit upwards. This results in the graph of the given function \(h(x) = \frac{1}{x}+1\). The asymptote along the x-axis is now translated to y=1. Consequently, the graph of \(h(x)\) is in the first quadrant (for x>0) and the second quadrant (for x<0), and it approaches but never touches the line y=1.

Key Concepts

Function Transformations

Function transformations allow us to modify basic functions into new forms by applying certain operations. These transformations can stretch, compress, reflect, or translate the graph of a function. In this context, the original function is \( f(x) = \frac{1}{x} \), which is a basic rational function.Types of Transformations:

• Translation: Moves the entire graph horizontally or vertically without altering its shape.
• Reflection: Flips the graph across a line, such as the x-axis or y-axis.
• Stretching/Compressing: Alters the graph's width or height by multiplying the function by a factor.

Transformation helps in graphing complex functions by starting with a fundamental form and applying changes. This means any new function can be analyzed based on how it departs from its basic function. For example, adding a constant shifts its position on the graph.

Graphing Functions

Graphing functions is the visual process of plotting a set of points that satisfies a given function. For the rational function \( h(x) = \frac{1}{x} + 1 \), graphing begins with its base function \( f(x) = \frac{1}{x} \). This function is known for its hyperbolic shape and key characteristics.Key Steps in Graphing:

• Identify the base function: Recognize the most basic form of the function you need to graph.
• Understand the behavior: For \( f(x) = \frac{1}{x} \), the graph approaches the x-axis and y-axis but never touches them. These lines are the asymptotes.
• Apply transformations: Modify the graph by applying any given transformations, such as shifts described by the equation.

Plotting the base graph first helps understand how transformations like vertical translation will affect the overall shape. This transformed graph maintains the asymptotes parallel to the original but moved according to changes.

Vertical Translation

Vertical translation is a specific type of function transformation where a function's graph is shifted up or down along the y-axis. It does not affect the shape or orientation but simply relocates the graph.Understanding Vertical Translation:Vertical translation changes the output of a function uniformly for all inputs. In the case of the function \( h(x) = \frac{1}{x} + 1 \), the addition of "+1" moves every point of \( f(x) = \frac{1}{x} \) up by one unit.Key Characteristics:

• Graph Movement: The entire graph shifts parallel to the y-axis.
• Asymptote Change: In the example, the horizontal asymptote shifts from y=0 to y=1.
• Affect on Intercepts: Vertical translations can affect intercepts if they exist, as it directly changes the y-values.

This kind of transformation is straightforward but crucial for correctly visualizing and interpreting the function's behavior in its new position.