Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. g(x)=1/(x+1) − 2

Verified step by step guidance

1. Identify the base function given, which is \( f(x) = \frac{1}{x} \). This is the parent function for the transformation.
2. Recognize the transformation inside the function's argument: \( x + 1 \). This represents a horizontal shift to the left by 1 unit because adding 1 inside the function moves the graph left.
3. Next, observe the \( -2 \) outside the fraction. This indicates a vertical shift downward by 2 units.
4. Combine these transformations to write the transformed function as \( g(x) = \frac{1}{x + 1} - 2 \). This means the graph of \( f(x) = \frac{1}{x} \) is shifted left 1 unit and down 2 units.
5. To graph \( g(x) \), start with the graph of \( f(x) = \frac{1}{x} \), shift every point left by 1, then shift every point down by 2. Also, update the vertical and horizontal asymptotes accordingly: the vertical asymptote moves from \( x=0 \) to \( x=-1 \), and the horizontal asymptote moves from \( y=0 \) to \( y=-2 \).

Key Concepts

Parent Rational Functions

The parent functions f(x) = 1/x and f(x) = 1/x² are basic rational functions with distinct shapes and asymptotes. Understanding their graphs, including vertical and horizontal asymptotes, is essential as they serve as the starting point for transformations.

Transformations of Functions

Transformations include shifts, stretches, and reflections applied to the parent function. For g(x) = 1/(x+1) - 2, the graph shifts left by 1 unit and down by 2 units, altering the position of asymptotes and the overall graph without changing its shape.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal asymptotes describe end behavior. Transformations shift these asymptotes accordingly, crucial for accurate graphing.