Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. h(x)=1/x² − 4

Verified step by step guidance

1. Identify the base function given, which is \(f(x) = \frac{1}{x^2}\). This is a rational function with a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\).
2. Look at the given function \(h(x) = \frac{1}{x^2} - 4\). Notice that it is the base function \(f(x)\) shifted vertically by subtracting 4.
3. Understand that subtracting 4 from \(f(x)\) shifts the entire graph downward by 4 units. This means the horizontal asymptote moves from \(y=0\) to \(y=-4\).
4. To graph \(h(x)\), start by sketching the graph of \(f(x) = \frac{1}{x^2}\), which has a vertical asymptote at \(x=0\) and approaches zero as \(x\) goes to infinity or negative infinity.
5. Then, shift every point on the graph of \(f(x)\) down by 4 units to get the graph of \(h(x)\). The vertical asymptote remains at \(x=0\), and the new horizontal asymptote is \(y=-4\).

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 shapes, asymptotes, and behavior helps in analyzing transformations applied to them.

Transformations of Functions

Transformations include shifts, stretches, and reflections applied to the parent function. For h(x) = 1/x² − 4, subtracting 4 shifts the graph downward by 4 units, affecting the position of the horizontal asymptote and the overall graph.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. For rational functions like h(x) = 1/x² − 4, vertical asymptotes occur where the denominator is zero, and horizontal asymptotes are determined by the end behavior, influenced by transformations.