Quadratic Equation Worksheet - Completing The Square Page 4

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GOAL
2
W
Q
F
V
F
RITING
UADRATIC
UNCTIONS IN
ERTEX
ORM
2
Given a quadratic function in standard form, y = ax
+ bx + c, you can use
2
completing the square to write the function in vertex form, y = a(x º h)
+ k.
Writing a Quadratic Function in Vertex Form
E X A M P L E 6
2
Write the quadratic function y = x
º 8x + 11 in vertex form. What is the vertex of
S
H
TUDENT
ELP
the function’s graph?
HOMEWORK HELP
Visit our Web site
S
OLUTION
2
for extra examples.
y = x
º 8x + 11
Write original function.
2
2
y +
?
= (x
º 8x +
?
) + 11
Prepare to complete the square for x
º 8x.
2
º8
2
}
2
y +
16
= (x
º 8x + 16) + 11
Add
= (º4)
= 16 to each side.
2
2
2
y + 16 = (x º 4)
+ 11
Write x
º 8x + 16 as a binomial squared.
2
y = (x º 4)
º 5
Solve for y.
2
The vertex form of the function is y = (x º 4)
º 5. The vertex is (4, º5).
Finding the Maximum Value of a Quadratic Function
E X A M P L E 7
Agriculture
The amount s (in pounds per acre) of sugar produced from sugarbeets can be
modeled by the function
2
s = º0.0655n
+ 7.855n + 5562
where n is the amount (in pounds per acre) of nitrogen fertilizer used. How much
fertilizer should you use to maximize sugar production? What is the maximum
amount of sugar you can produce?
Source: Sugarbeet Research and Education Board of Minnesota and North Dakota
S
OLUTION
The optimal amount of fertilizer and the maximum amount of sugar are the
coordinates of the vertex of the function’s graph. One way to find the vertex is
to write the function in vertex form.
2
s = º0.0655n
+ 7.855n + 5562
2
s = º0.0655(n
º 120n) + 5562
2
s º 0.0655(
?
) = º0.0655(n
º 120n +
?
) + 5562
2
s º 0.0655(3600) = º0.0655(n
º 120n + 3600) + 5562
2
s º 236 = º0.0655(n º 60)
+ 5562
2
s = º0.0655(n º 60)
+ 5798
The vertex is approximately (60, 5798). To maximize sugar production, you
should use about 60 pounds per acre of nitrogen fertilizer. The maximum amount
of sugar you can produce is about 5800 pounds per acre.
5.5 Completing the Square
285

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