1. Definitions
   • Quadratic Function

     a function of the form

     

     where a, b, and c are real numbers, with .

   • parabola

     the graph of a quadratic function. It looks like either an umbrella or a bowl. The max or the min is called the vertex, and it occurs at the axis of symmetry.

     The vertex occurs at the point

     

     The graph opens upward (looks like an umbrella) if a>0, and opens downward (looks like an umbrella) if a<0.

2. Theorems/Formulas
   • quadratic formula

     the roots (x-intercepts, places where the function is zero) of a quadratic function are given by the formula

     

3. Properties/Tricks/Hints/Etc.

   Let f be any function and let h and k be positive constants. The graph of

   • y=f(x)+k is the graph of f translated upward by an amount k.

   • y=f(x)-k is the graph of f translated downward by an amount k.

   • y=f(x+h) is the graph of f translated left by an amount h.

   • y=f(x-h) is the graph of f translated right by an amount h.

   • y=-f(x) is the graph of f reflected across the x-axis.

   • y=f(-x) is the graph of f reflected across the y-axis.

   Note that these are examples of compositions of functions.

4. Summary

   This section introduces the quadratic function in more detail, and shows how various aspects of the graph of a quadratic (a parabola) are related to the formula : the vertex, the roots, the axis of symmetry, etc.

   General rules for translations and reflections are provided, along with some examples.