In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. h(x)=1/(x−3)²+1

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 horizontal shift inside the function: the expression $x-3$ inside the denominator indicates a shift of the graph 3 units to the right.
3. Understand the vertical shift outside the function: the +1 outside the fraction means the entire graph is shifted 1 unit upward.
4. Combine the transformations: start with the graph of , shift it 3 units to the right to get , then shift the graph 1 unit up to get .
5. Use these transformations to sketch the graph: note the vertical asymptote at $x=3$ due to the denominator, and the horizontal asymptote at $y=1$ because of the vertical shift.

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, helps in visualizing how transformations affect the function's graph.

Transformations of Functions

Transformations include shifts, stretches, and reflections applied to the parent function. For h(x) = 1/(x−3)² + 1, the graph shifts right by 3 units and up by 1 unit, altering the position of asymptotes and the overall shape without changing the basic form.

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. Identifying these helps in sketching accurate graphs of rational functions.