Absolute Value & Rational Functions
Absolute value functions create V-shaped graphs that never go below (or above) their vertex. The general form f(x) = a|x - h| + k tells you everything: if a > 0, it opens upward; if a < 0, it opens downward. The vertex sits at (h, k).
These functions always have a domain of all real numbers, but their range depends on which way they open. Upward-opening functions have range k,+∞),whiledownward−openingoneshaverange(−∞,k.
Rational functions are fractions with polynomials on top and bottom. They're famous for their asymptotes - invisible lines the graph approaches but never touches. Vertical asymptotes occur where the denominator equals zero, while horizontal asymptotes depend on the degrees of the numerator and denominator.
Key Insight: Rational functions are undefined wherever their denominator equals zero - these points create vertical asymptotes that split the graph into separate pieces.