91Ó°ÊÓ

A toolkit function is a basic function that forms a building block for more complex functions. In this case, our toolkit function is the reciprocal function, represented by the equation \( y = \frac{1}{x} \). This function is fundamental in understanding rational functions.
The graph of \( y = \frac{1}{x} \) is a hyperbola that resides in two separate quadrants. It features vertical and horizontal asymptotes on the x-axis (\( x = 0 \)) and y-axis (\( y = 0 \)), respectively. These asymptotes are invisible barriers that the curve approaches but never quite touches.
When graphing rational functions, starting with a toolkit function allows for easier visualization and understanding of transformations.

Transformations involve altering the graph of a function in various ways, such as shifting, reflecting, stretching, or compressing. They are vital in graphing because they help display how the basic shape of a function evolves as further changes are applied.

• Shifting involves moving the graph in a specific direction without altering its shape.
• Stretching or compressing involves changing the size of the graph, either making it steeper or more flattened.

For the function \( f(x) = \frac{1}{x+2} - 1 \), the primary transformations are horizontal and vertical translations.

Horizontal translations move a graph left or right along the x-axis. They occur by changing the input variable \( x \) within the function.
In the function \( f(x) = \frac{1}{x+2} - 1 \), notice the term \( x+2 \) inside the reciprocal part. This implies a horizontal shift. A positive number inside the term moves the graph to the left, while a negative number moves it to the right.
Therefore, \( x+2 \) indicates a shift to the left by 2 units from the original position. Consequently, for our graph, the vertical asymptote that was originally at \( x = 0 \) in the toolkit function \( y = \frac{1}{x} \) will now move to \( x = -2 \).

Vertical translations lift or lower a graph along the y-axis without altering its horizontal positioning. This change is made by adding or subtracting a numerical value in the function expression.
In the function \( f(x) = \frac{1}{x+2} - 1 \), you see a \(-1\) outside the reciprocal term. This signals a vertical translation, moving the entire graph down by 1 unit.
As a result, the horizontal asymptote originally located at \( y = 0 \) for the toolkit function \( y = \frac{1}{x} \) is translated downwards to \( y = -1 \). These shifts help in envisioning the final graph of the given function.