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MAT122 :: Lecture Note :: Week 12
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Assignment(s)

Quote of the Week {Furman.edu:: Mathematical Quotations Server}
I have learned, that if one advances confidently in the direction
of his dreams, and endeavors to live the life he has imagined, he
will meet with a success unexpected in common hours.
-- Henry David Thoreau (01817-01862) {American author; more...} [life/dream] [log]
Road Signs of the Week {NASA.gov:: Astronomy Picture of the Day}
near I8/I10 jct AZ | near Sandy OR | near Hannibal MO [log]

Primer: Quadratic Equations and Functions

Quadratic equations are 2nd-degree polynomials.

Quadratic equation in standard form: ax^2 + bx + c = 0

Quadratic function in standard form: f(x) = ax^2 + bx + c 

   ax^2 + bx + c = 0          [2nd-degree polynomial (tri-nomial)]
   where a, b, c are coefficients and a ≠ 0
   [if a = 0, then it's a linear equation]

   a is the quadratic coefficient 
   b is the linear coefficient 
   c is the constant coefficient

Quadratic equations can always be solved by using the quadratic formula. 

   x = (-b +- sqrt(b^2 - 4ac)) / (2a)

   b^2 - 4ac  is called the discriminant (d)
   if d > 0, then two distinct real roots
   if d = 0, then one distinct real root
   if d < 0, then two distince complex roots

The graph of a quadratic function is a parabola. 

   f(x) = ax2 + bx + c

   a > 0  implies the parabola opens up
   a < 0  implies the parabola opens down

   vertical intercept:  (0, c)

   when |a| increases, the parabola becomes narrower
   when |a| decreases, the parabola becomes wider

   vertex  is the high (a < 0) and low point (a > 0) of a parabola 
   vertex  is sometimes called the inflection point

   x-coordinate of the vertex:  x = -b/(2a)
          vertex ordered-pair:  (x, f(x)) or (-b/(2a), f(-b/(2a)))
   equation for the axis-of-symmetry:  x = -b/(2a)

   domain:  (-INF, INF)
    range:  if vertex is (h, k) 
            [k, INF) when a > 0   and   (-INF, k] when a < 0

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