Describing the graph of a
Parabola
There are four different ways to describe the graph of a
parabola:
1. It intersects the x-axis twice.
2. It is tangent to the x-axis. (It only intersects it once)
3. It lies entirely above the x-axis.
4. It lies entirely below the x-axis.
We are going to explore all of these ways by looking at
the discriminant and the coefficient of the squared
term.
We need to be able to
determine if a parabola opens
up or down. To do that, we
look at the coefficient of the
squared term.
Now type the following
equation into y= on your
calculator.
y  3  4x  x
2
Type in the following
equation into y= on your
calculator.
y  x  6x  5
2
What did we learn?
If the squared term is positive, it opens up. If
it is negative, it opens down!
We need to be able to
determine what makes a
parabola touch the x-axis and
lie above or below it:
If the roots are real, the
parabola touches the x-axis:
Ex:
a 1
b5
c3
The discriminant is a
positive, non-perfect square.
Therefore the roots are:
1. Real
2. Irrational
3. Unequal
y  x  5x  3
2
b  4 ac
2
5 2  4 13 
13
We would describe this graph
as touching the x-axis twice!
If the roots are imaginary, the
parabola doesn’t touch the x-axis:
y   x  5x  8
2
a  1
b5
c  8
Opens
down
b  4 ac
2
5 2  4  1 8 
7
The discriminant is negative.
Therefore the roots are:
1. Imaginary
We would describe this graph
as follows:
It lies entirely below the xaxis.
If the roots are imaginary and the coefficient of the squared term is
negative, it lies entirely below the x-axis. If the roots are imaginary and
the squared term is positive, it lies entirely above the x-axis.
Now lets determine what the graph
looks like.
#13:
a 1
b  2
c  8
y  x  2x  8
2
Page 2
#16:
a 1
b0
Parabola opens up
b  4 ac
2
 2 2  4 1 8 
36
The discriminant is a
positive, perfect square.
Therefore the roots are:
1. Real
2. Rational
3. Unequal
Since the roots are unequal and real,
the graph intersects the x-axis twice!
c  6
y  x 6
2
Parabola opens up
b  4 ac
2
0 2  4 1  6 
24
The discriminant is a
positive, non-perfect square.
Therefore the roots are:
1. Real
2. Irrational
3. Unequal
Since the roots are unequal and real,
the graph intersects the x-axis twice!
#19:
a  10
b0
c0
y  10 x
2
Parabola opens up
b  4 ac
2
0 
2
 4 10 0 
0
#20: y   x  3 x  7
2
a  1
b3
c  7
Page 2
Parabola opens down
b  4 ac
2
3 2  4  1 7 
 19
The discriminant is zero.
Therefore the roots are:
1. Real
2. Rational
3. Equal
The discriminant is negative.
Therefore the roots are:
1. Imaginary
Since the roots are equal the
parabola is tangent to the x-axis.
Since the roots are imaginary and the
parabola opens down, then the
parabola lies entirely below the xaxis.
#34: The roots of x 2  2 x  k  0 are equal when k is:
Remember, the roots are equal
when the discriminant is 0.
x  2x  k  0
2
a 1
b  4 ac  0
b2
 2 2  4 1 k   0
ck
4  4k  0
2
 4k
4  4k
4
4
1 k
 4k
Page 2
#36: The roots of x 2  bx  8  0 are imaginary when b is:
Page 2
Remember, the roots are imaginary
when the discriminant is negative.
x  bx  8  0
2
a 1
b  4 ac  0
bb
b 2  4 18   0
c8
b  32  0
2
< 0 means negative!
2
Since this is multiple choice, sub in
values from choices until one works!
 5 2  32
0
25  32  0
70
#38: If the graph of y  ax 2  bx  c is tangent to the xaxis, then the roots of y  ax 2  bx  c are:
If a graph is tangent to the x-axis, it
ONCE
only touches the x-axis__________.
The roots are:
Real
Rational
Equal
Page 2
#36: The roots of x 2  bx  16  0 are equal when b is:
Page 2
Remember, the roots are equal
when the discriminant is zero.
x  bx  16  0
2
a 1
b  4 ac  0
bb
b 2  4 116   0
c  16
b  64  0
b  8 b  8   0
2
2
b8  0
b 8
b8 0
b  8
= 0 means equal roots.
Find the largest integral value of k
such that the roots of the given
equation are real.
# 42 : x  6 x  k  0
2
a 1
b  4 ac  0
2
b6
ck
 means
real.
6 2  4 1 k   0
36  4 k  0
 4k  4k
36  4 k
4
4
9 k
k 9
Homework
•Page 2
#15,18,21,35,41,
43,46