Follow the line from both sides; if they meet at the same height, the limit exists. Algebraic View: Try direct substitution first; if you get , factor or rationalize.
Compare the degrees of the numerator and denominator. 4. Trigonometry via the Unit Circle Coordinates: Remember
If a graph looks weird, plot 3-5 specific points to anchor it. Precalculus with Limits: A Graphing Approach
Always check for "illegal" math (denominators of zero or negatives in square roots).
Graphing is easier when you view equations as "shifts" of the parent functions. Horizontal Shifts: (Right) or Reflections: (Over x-axis) or (Over y-axis) Scaling: stretches or shrinks the graph vertically. 3. Analyze Polynomial & Rational Functions Follow the line from both sides; if they
Use the Leading Coefficient Test to see where the graph goes as
Find x-intercepts and determine "multiplicity" (does the graph cross or bounce?). Asymptotes: Vertical: Where the denominator equals zero. Graphing is easier when you view equations as
Identify Amplitude (height), Period (length of one cycle), and Phase Shift (horizontal slide). Identities: Use Pythagorean identities ( ) to simplify expressions before graphing. 5. Limits: The Bridge to Calculus