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

Verified step by step guidance

1. Identify the base function. Here, the base function is \(f(x) = \frac{1}{x^2}\), which has a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\).
2. Analyze the transformation inside the function's denominator. The function is \(g(x) = \frac{1}{(x+2)^2} - 1\), so the \(x\) is replaced by \(x+2\). This means the graph shifts horizontally to the left by 2 units.
3. Consider the vertical shift. The \(-1\) outside the fraction means the entire graph shifts downward by 1 unit, moving the horizontal asymptote from \(y=0\) to \(y=-1\).
4. Determine the new vertical asymptote. Since the denominator is zero when \(x+2=0\), the vertical asymptote shifts from \(x=0\) to \(x=-2\).
5. Summarize the transformations: start with \(f(x) = \frac{1}{x^2}\), shift left 2 units to get \(\frac{1}{(x+2)^2}\), then shift down 1 unit to get \(g(x) = \frac{1}{(x+2)^2} - 1\). Use these to sketch the graph with the new asymptotes and shape.

Key Concepts

Parent Rational Functions

Parent rational functions like f(x) = 1/x and f(x) = 1/x^2 serve as the basic models for graphing more complex rational functions. Understanding their shapes, asymptotes, and behavior helps in applying transformations to graph related functions.

Transformations of Functions

Transformations include shifts, reflections, stretches, and compressions applied to parent functions. For example, g(x) = 1/(x + 2)^2 - 1 involves a horizontal shift left by 2 units and a vertical shift down by 1 unit, altering the graph's position 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. For g(x), the vertical asymptote is at x = -2, and the horizontal asymptote is y = -1.