MAC1105: Horizontal Intercepts and Related Topics
The points at which a graph intersects the horizontal axis are called the horizontal intercepts of the graph, and are the
real number inputs that result in outputs of zero. These inputs are called
zeros
of the function.
For example, the
linear function
f(x)
2x
10
=

has one zero at
x
5
=
, which means that
f(5)
0
=
and the point
(5,0)
is the
x
intercept of the line
y
2x
10
=

=
.
Algebraically we compute intercepts by letting the other variable in the formula
equal
0
.
When using the formula
y
2x
10
=

, letting
x
equal
0
locates the
y

intercept at
10
, and letting
y
equal
0
and solving the equation
2x
10
0

=
=
locates the
x
intercept at
5
.
This last computation is relatively easy because the
equation
2x
10
0

=
is linear.
For other types of functions this can be a much
more challenging process.
We have been examining quadratic functions whose graphs are called
parabolas
.
If the vertex is also the
x

intercept, then the parabola will have only that point as a single
x
intercept.
But if the vertex is not on the
x
axis, the
parabola will have either two or no
x
intercepts depending upon whether the arms of the parabola open toward or
away from the
x
axis.
Finding these
x
intercepts amounts to solving the quadratic equation
2
ax
bx
c
0
+
+
=
+
, and there are various
methods for doing so.
Algebraic methods include
factoring
and the
quadratic formula
.
A graphing calculator can
also be used, but will not find exact values if the intercepts occur at irrational numbers.
Consider the quadratic equation
2
x
2x
15
0


=

=
.
This equation can be solved by factoring the quadratic and
setting each factor equal to
0
.
2
x
2x
15
0
(x
3)(x
5)
0
x
3
0 x
5
0
x
3
x
5


=


=

=

=
+

=
+

=


+
=

=
=
= 
=
= 
=
So the
x
intercepts are at
3
and
5
.
However, if the constant term is changed from
15
to
16
, the quadratic expression
2
x
2x
16




no longer factors.
In this case we can use the quadratic formula
2
b
b
4ac
x
2a

±

=
.
(See page 4 of this handout.
Substituting
into this formula
(a
1, b
2, c
16)
=
= 
= 
results in the computation:
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(
)
2
( 2)
( 2)
4(1)( 16)
x
2(1)
2
4
64
2
68
2
4 17
2
2
2
2 1
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 Fall '08
 Algebra, Quadratic Formula, Quadratic equation

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