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

Verified step by step guidance

1. Identify the base function given, which is \( f(x) = \frac{1}{x^2} \). This function has a vertical asymptote at \( x=0 \) and a horizontal asymptote at \( y=0 \).
2. Recognize that the function \( g(x) = \frac{1}{(x+2)^2} \) is a transformation of \( f(x) = \frac{1}{x^2} \) involving a horizontal shift.
3. Determine the horizontal shift by setting the inside of the denominator equal to zero: \( x + 2 = 0 \) which gives \( x = -2 \). This means the graph of \( f(x) \) shifts 2 units to the left.
4. Update the vertical asymptote from \( x=0 \) to \( x = -2 \) because the denominator is zero at \( x = -2 \). The horizontal asymptote remains \( y=0 \) since the degree of the numerator is less than the degree of the denominator.
5. Sketch the graph by shifting the original \( f(x) = \frac{1}{x^2} \) curve 2 units to the left, keeping the shape the same, with the vertical asymptote at \( x = -2 \) and the horizontal asymptote at \( y=0 \).

Key Concepts

Parent Rational Functions

Parent rational functions like f(x) = 1/x and f(x) = 1/x² serve as the basic models for more complex rational functions. Understanding their graphs, including asymptotes and general shape, is essential before applying transformations. For example, f(x) = 1/x has vertical and horizontal asymptotes at x=0 and y=0, respectively.

Transformations of Functions

Transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For g(x) = 1/(x+2)², the '+2' inside the denominator shifts the graph horizontally left by 2 units. Recognizing how changes inside the function affect the graph helps in sketching the transformed function accurately.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, indicating undefined points, while horizontal asymptotes describe end behavior as x approaches infinity. For g(x) = 1/(x+2)², the vertical asymptote is at x = -2, and the horizontal asymptote is y = 0.