Quadratic Polynomials
If a > 0 then the graph of ax2 is obtained by starting with the graph of x2 ,
and then stretching or shrinking vertically by a.
If a < 0 then the graph of ax2 is obtained by starting with the graph of x2 ,
then flipping it over the x-axis, and then stretching or shrinking vertically by
the positive number a.
When a > 0 we say that the graph of ax2 “opens up”. When a < 0 we say
that the graph of ax2 “opens down”.
I Cit
i-a
x-ax~S
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2
IfIf a,
c,
dd 2⇥ R
and
aa 6=
0,
then
the
graph
of
a(x
+
c)
2 + d is obtained by
a,
c,
R
and
=
⇤
0,
then
the
graph
of
a(x
+
c)
2 + d is obtained by
If
a,
c,
d
⇥
R
and
a
=
⇤
2 0, then the graph of a(x + c) + d is obtained by
shifting
the
graph
of
ax
horizontally
shifting the
the graph
graph of
of ax
ax22 horizontally
horizontally by
by c,
c, and
and vertically
vertically by
by d.
d. (Remember
(Remember
shifting
by
c,
and
vertically
by
d.
(Remember
that
d
>
0
means
moving
up,
d
<
0
means
moving
down,
c
>
0
means
that dd >
> 00 means
means moving
moving up,
up, dd <
< 00 means
means moving
moving down,
down, cc >
> 00 means
means moving
moving
that
moving
left,
and
c
<
0
means
moving
right.)
left, and
and cc <
< 00 means
means moving
moving right.)
right.)
left,
2
IfIf aa 6=
0,
the
graph
of
aa function
ff(x)
=
a(x
+
c)
2 + d is called a parabola.
=
⇤
0,
the
graph
of
function
(x)
=
a(x
+
c)
2 + d is called a parabola.
2 a function f (x) = a(x + c) + d is called a parabola.
If
a
=
⇤
0,
the
graph
of
The
point
c,
d)
2⇥ R
isis called
The point
point ((( c,
c,d)
d) ⇥
R22 is
called the
the vertex
vertex of
of the
the parabola.
parabola.
The
R
called
the
vertex
of
the
parabola.
2
Example.
Below
isis the
parabola
that
isis the
graph
of
10(x
+
2)
2
Example.
Below
the
parabola
that
the
graph
of
10(x
+
2)
2
Example.
Below
is
the
parabola
that
is
the
graph
of
10(x
+
2)
vertex
isis (( 2,
3).
vertex is
2, 3).
3).
vertex
( 2,
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2
2
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3.
3. Its
Its
3.
Its
A quadratic polynomial is a degree 2 polynomial. In other words, a quadratic polynomial is any polynomial of the form
p(x) = ax2 + bx + c
where a, b, c 2 R and a 6= 0.
Completing the square
You should memorize this equation: (it’s called completing the square.)
b ⌘2
ax + bx + c = a x +
+c
2a
2
⇣
b2
4a
Let’s check that the equation is true:
⇣
b ⌘2
a x+
+c
2a
⇣
h b i2 ⌘
b2
b
2
= a x + 2x +
+c
4a
2a
2a
h b i2
b
2
= ax + a2x + a
+c
2a
2a
h b2 i b2
2
= ax + bx + a
+c
4a2
4a
b2
b2
2
= ax + bx +
+c
4a 4a
= ax2 + bx + c
b2
4a
b2
4a
Graphing quadratics
Complete the square to graph quadratic polynomials: If p(x) = ax2 +bx+c,
b 2
b2
then p(x) = a x + 2a
+ c 4a
. Therefore, the graph of p(x) = ax2 + bx + c
b
is a parabola obtained by shifting the graph of ax2 horizontally by 2a
, and
2
b
vertically by c 4a
. The parabola opens up if a > 0 and opens down if a < 0.
Example. To graph 3x2 + 5x 2, complete the square to find that
1
3x2 + 5x 2 equals 3(x 56 )2 + 12
. To graph this polynomial, we start
2
with the parabola for 3x , which opens down since 3 is negative.
150
.
.
5
11
2 2
Shift
. .The
Shift the
the parabola
parabola for
for 3x3xright
rightbyby6 56and
andthen
thenup
upby
by1212
Theresult
resultisisthe
the
22
22
5
1
2
graph
for
3x
+
5x
2.
Notice
that
the
graph
looks
like
the
graph
of
3x
,,
Shiftfor
the parabola
rightthat
by 6the
and
thenlooks
up by
The
result
the
graph
3x + 5x for2.3x
Notice
graph
like
graph
of is 3x
12 .the
5
1
2
1). graph looks like the graph of
except
that
its
point
((65,,the
graph
+
5x isis
2.the
Notice
that
3x2 ,
exceptfor
that3x
its vertex
vertex
the
point
).
12
6
12
1
except that its vertex is the point ( 56 , 12
).
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Discriminant
The discriminant of ax2 + bx + c is defined to be the number b2
4ac.
How many roots?
If p(x) = ax2 + bx + c, then the following chart shows how the discriminant
of p(x) determines how many roots p(x) has:
b2
4ac
number of roots
>0
2
=0
1
<0
0
Example. Suppose p(x) = 2x2 + 3x 1. Because 32
9 8 = 1 is positive, p(x) = 2x2 + 3x 1 has two roots.
4( 2)( 1) =
Why the discriminant of a quadratic polynomial tells us about the number
of roots of the polynomial, and why the information from the above chart is
true, will be explained in lectures.
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Finding roots
If ax2 + bx + c has at least one root – which is the same as saying that
b2 4ac 0 – then there is a formula that tells us what those roots are.
Quadratic formula
If p(x) = ax2 + bx + c with a 6= 0 and if b2
then the roots of p(x) are
b+
p
b2
2a
4ac
and
b
p
b2
2a
4ac
0,
4ac
Notice in the quadratic formula, that we need a 6= 0 to make sure that we
are not dividing by 0, and we need b2 4ac 0 to make sure that we aren’t
taking the square root of a negative number.
Also recall that if ax2 + bx + c has only one root, then b2 4ac = 0. That
means the two roots from the quadratic formula are really the same root.
It’s a good exercise in algebra to check that the quadratic equation is true.
To check that it’s true, you need to check that
⇣ b + pb2 4ac ⌘
p
=0
2a
and that
⇣ b pb2 4ac ⌘
p
=0
2a
Example. We checked above that p(x) = 2x2 +3x 1 has 2 roots, because
its discriminant equalled 1. The quadratic formula tells us that those roots
equal
p
3+ 1
3+1
2 1
=
=
=
2( 2)
4
4 2
and
p
3
1
4
=
=1
2( 2)
4
153
Exercises
For each of the quadratic polynomials in problems #1-6:
• Complete the square.
⇣
2
That means rewrite ax + bx + c as a x +
b
2a
⌘2
+c
b2
4a .
• What’s the vertex of the corresponding parabola?
The vertex of the parabola for a(x + c)2 + d is the point ( c, d).
• Is its parabola opening up, or opening down?
The parabola opens up if the leading coefficient of the quadratic
polynomial is positive. The parabola opens down if the leading
coefficient is negative.
• What’s its discriminant?
The discriminant of ax2 + bx + c is b2 4ac.
• How many roots does it have?
There are two roots if the discriminant is positive, one root if the
discriminant equals 0, and zero roots if the discriminant is negative.
• What are its roots (if it has any)?
p
2
p
2
If ax2 + bx + c has roots, they are b+ 2ab 4ac and b 2ab 4ac .
• Match its graph with one of the lettered graphs on the next page.
1.)
2x2
2x + 12
2.) x2 + 2x + 1
3.) 3x2
4.)
9x + 6
4x2 + 16x
5.) x2 + 2x
19
1
6.) 3x2 + 6x + 5
154
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7.) Suppose you shoot a feather straight up into the air, and that t is the
time measured in seconds that follow after you shoot the feather into the air.
If the height of the feather at time t is given by 2t2 + 20t feet, then what
is the maximum height that the feather reaches? How many seconds does it
take for the feather to reach its maximum height?
8.) Let’s say you make cogs for a living. After accounting for the cost of
building materials, you earn a profit of x2 10x + 45 cents on the x-th cog
that you make. Which cog do you earn the least amount of profit for making?
How much profit do you earn for that cog?
Solve for x in the equations given in #9-11.
9.)
x2 + x + 1
=1
2x + 1
10.)
x2 + 3
=1
x2 + 2x + 5
156
11.)
3x 5
=
x2
2