Calculus 2nd Edition Rogawski

18 CHAPTER 1 PRECALCULUS REVIEW
The technique of completing the square consists of writing a quadratic polynomial
as a multiple of a square plus a constant:
) 2
(
ax 2 + bx + c = a x + b
} {{
2a
}
Square term
+
4ac − b2
}
4a
{{ }
Constant
2
It is not necessary to memorize this formula, but you should know how to carry out the
process of completing the square.
Cuneiform texts written on clay tablets
show that the method of completing the
square was known to ancient Babylonian
mathematicians who lived some 4000
years ago.
EXAMPLE 4 Completing the Square Complete the square for the quadratic polynomial
4x 2 − 12x + 3.
Solution First factor out the leading coefficient:
(
4x 2 − 12x + 3 = 4 x 2 − 3x + 3 )
4
Then complete the square for the term x 2 − 3x:
(
x 2 + bx = x + b 2
Therefore,
4x 2 − 12x + 3 = 4
) 2
− b2
4 , x2 − 3x =
( (
x − 3 2
) 2
− 9 4 + 3 4)
(
x − 3 ) 2
− 9 2 4
(
= 4 x −
2) 3 2
− 6
The method of completing the square can be used to find the minimum or maximum
value of a quadratic function.
y
9
5
x
2
FIGURE 10 Graph of f (x) = x 2 − 4x + 9.
EXAMPLE 5 Finding the Minimum of a Quadratic Function Complete the square and
find the minimum value of f (x) = x 2 − 4x + 9.
Solution We have
f (x) = x 2 − 4x + 9 = (x − 2) 2 − 4 + 9 =
This term is ≥ 0
{ }} {
(x − 2) 2 + 5
Thus, f (x) ≥ 5 for all x, and the minimum value of f (x) is f(2) = 5 (Figure 10).
1.2 SUMMARY
• A linear function is a function of the form f (x) = mx + b.
• The general equation of a line is ax + by = c. The line y = c is horizontal and x = c
is vertical.
• Three convenient ways of writing the equation of a nonvertical line:
– Slope-intercept form: y = mx + b (slope m and y-intercept b)
– Point-slope form: y − b = m(x − a) [slope m, passes through (a, b)]
– Point-point form: The line through two points P = (a 1 ,b 1 ) and Q = (a 2 ,b 2 ) has
slope m = b 2 − b 1
a 2 − a 1
and equation y − b 1 = m(x − a 1 ).
• Two lines of slopes m 1 and m 2 are parallel if and only if m 1 = m 2 , and they are perpendicular
if and only if m 1 = −1/m 2 .