Greenwood press

QUADRATIC FUNCTIONS
89
Graph of the quadratic function
f (x) = −3x
2
+ 2x + 5.

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Rewrite the expanded trinomial as a perfect square and simplify:
f (x) = a(x +
b
2a
)
2
−
b
2
4a
+ c.
Compare this expression to the vertex form of a quadratic function and notice 
that the vertex can be represented as 
(−
b
2a
, −
b
2
4a
+ c). This coordinate will serve 
as a shortcut to find the highest point when a < 0, and lowest point when a > 0,
in an application that uses quadratic functions.
Take, for instance, a business setting that sells sport memorabilia. The
demand, and hence the price, for a baseball player’s autographed ball may
decline as more of them become available. Suppose that the price of an auto-
graphed ball, 
a, from a new hall-of-famer begins at $200 and declines by five
cents, or $0.05, for every ball, 
x, sold. This relationship would be represented by
the equation 
a = 200 − 0.05x. The revenue, r, obtained from selling the balls
would be the product of the number of balls sold and the price for each ball, or
r = x(200 − 0.05x) = 200x − 0.05x
2
. The business owner will have to pay for
general start-up costs such as hiring the baseball player to sign autographs and
renting a place to sell the merchandise, as well as paying for the materials, such
as the cost of each ball. Suppose the start-up costs are $1,300 and the business
owner pays $1.25 for each new ball. Then the cost, 
c, that the business assumes
in terms of the number of balls sold will be 
c = 1.25b + 1300.
The profit, 
p, is the difference between the revenue and cost, or r − c, which
equals
(200x − 0.05x
2
) − (1.25x + 1300), and simplifies to p = -0.05x
2
+
199.75x − 1300. In a quadratic function in the form of f(x) = ax
2
+ bx + c, a
maximum value will occur when 
x = –
b
2a
, since a < 0. In this case, a maximum 
profit will occur when approximately 1,997 balls are sold
(x = −
199.75
2(
−0.05)
=
1997.5). In that case, a reasonable sale price of the “limited edition” ball should
be around 
200 − 0.05(1997) = $100.15. Although, to appease the human psy-
che, a more reasonable price might be twenty cents cheaper at $99.95 so that con-
sumers feel like they are getting a deal by paying less than $100. A graph and
table of values can also support this sale price as a means of producing almost a
maximum possible profit.
The vertical height, 
h, of an object is determined by the quadratic equation
h = –0.5gt
2
+ v
o
t + h
o
, where 
g is the acceleration due to earth’s gravity (9.8
m/sec
2
),
v
o
is the initial vertical velocity, and 
h
o
is the initial height of the object.
Therefore an object with an initial vertical velocity of 45 meters per second,
thrown at a height of 0.4 meters, can be modeled with the equation 
h = –4.9t
2
+
45t + 0.4. Engineers of fireworks can use this type of function so that the rockets
explode at a time where optimal height offers safety as well as viewing pleasure. 
This quadratic equation can also be used to measure the initial vertical veloc-
ity of an object thrown in the air, such as a ball, assuming that it reaches the
ground with minimal air resistance. For example, if a ball thrown at a height of
1.45 meters is airborne for 3.84 seconds, then the values can be substituted into
the equation 
h = –0.5gt
2
+ v
o
t + h
o
to solve for 
v
o
. In this case, the height after
3.84 seconds is equal to 0 meters, because that is the amount of time it takes for