If a, b, c are real numbers with a not equal to zero , then
the function f (x) = ax^2 + bx + c is a
quadratic function and its graph is a parabola.
· Axis of symmetry: The axis of symmetry is a
vertical line :
· Vertex of a parabola : The vertex of a parabola is
* The x- coordinate is the same as the axis of
symmetry
* The y-coordinate is found by plugging the
x-coordinate of the vertex into the original function.
· x- intercepts :
( Using the calculator ) For each x-intercept: 
(CALC) 2:zero
( Algebraically ) Let y = 0 and solve for x .
· y-intercept:
(Using the calculator) 
then
then 
(Algebraically) Let x = 0 and solve for y.
· Maximum/Minimum:
(Using the calculator) 
(CALC) 3:minimum OR
4:maximum
If a > 0, then there is a minimum.
If a < 0, then there is a maximum.
II. Vertex Form of a Quadratic Function
The vertex form of a parabola is f (x) = a(x - h)^2 + k
, where the vertex is (h, k). The axis of
symmetry is h = k.
Example: Given f (x) = x^2 - 4,
a) Graph the function.
b) Find the vertex.
c) Label the axis of symmetry.
d) Find the x- and y-intercepts.
e) Find max/ min value .
f) State the range.
Applications
Example 1: According to data from the U.S. Census
Bureau, the population of Cleveland, Ohio
(in thousands), in year x can be approximated by g(x) = -0.172x^2 + 16.82x +
487.62, where x = 0
corresponds to 1900. In what year did Cleveland have its largest population?
Example 2: The marketing research department for a
company that manufactures and sells
“notebook” computers established the fol lowing revenue and cost functions:
R(x) = x(2,000 – 60x) and C(x) = 4,000 + 500x
where x is thousands of computers and C(x) and R(x) are in
thousands of dollars. Both have
domain 1 ≤ x ≤ 25.
a) Graph both functions on your calculator.
b) Find the break-even points.
c) For what outputs will a loss occur?
d) For what outputs will a profit occur?
Example 3: The number of women employed full-time
in civilian jobs has increased
dramatically in the past century. The following table shows the number of
employed women
(in millions) in selected years between 1910 and 2003.
Using quadratic regression on a graphing calculator, find
the quadratic function that best fits the
data, where x is the number of years since 1900 and y is the number of
employed women, in
millions.
