In Exercises 45–56, 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: here, the base function is $f\left(x\right)=\frac{1}{x^2}$, which is the reciprocal squared function.
2. Recognize the transformation inside the function: the function given is $g\left(x\right)=\frac{1}{{(}^x}$. The $x+\; 2$ inside the denominator indicates a horizontal shift.
3. Determine the direction of the horizontal shift: since it is $x+\; 2$, the graph of $f(x)$ shifts 2 units to the left.
4. Apply the horizontal shift to the base graph: take the graph of $f(x)\; =\; 1/x^2$ and move every point 2 units left to get the graph of $g(x)$.
5. Note the shape and asymptotes: the vertical asymptote moves from $x=0$ to $x=-2$, and the horizontal asymptote remains 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 graphing more complex rational functions. Understanding their shapes, asymptotes, and behavior helps in visualizing transformations applied to these functions.

Transformations of Functions

Transformations include shifts, stretches, and reflections applied to the parent function. For g(x) = 1/(x+2)², the (x+2) inside the denominator indicates a horizontal shift of the graph 2 units to the left.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. For rational functions like g(x), vertical asymptotes occur where the denominator is zero (x = -2), and horizontal asymptotes describe end behavior, often y = 0 for these functions.