Roots (intercepts)
Let's first solve this parabola to obtain the roots.
Factorise
x
2
−
4
x
+
3
:
(
x
−
3
)
(
x
−
1
)
x
−
3
=
0
,
x
−
1
=
0
x
=
3
,
x
=
1
Therefore, the roots are
x
=
3
and
x
=
1
Vertex
The vertex can be found by using the formula
−
b
2
a
. In our case,
a
=
1
,
b
=
−
4
Substitute
a
=
1
,
b
=
−
4
into the formula:
−
−
4
(
2
)
(
1
)
−
(
−
2
)
2
Therefore, the
x
coordinate of the vertex is at
x
=
2
*Note that since we already knew the two roots, we could have easily gotten the
x
coordinate of the vertex by taking the average of the two roots. In this case, it is
3
+
1
2
=
2
. I solved it by this formula because it is important to know as well.
Substitute
2
into the original equation:
2
2
−
(
4
)
(
2
)
+
3
4
−
8
+
3
−
1
Therefore, the
y
coordinate of the vertex is at
y
=
−
1
Then you can roughly sketch out the graph using the two roots and the vertex.
Here is the graph:
graph{x^2-4x+3 [-10, 10, -5, 5]}
For better accuracy, you can always make a table of values. But having 3 points is usually enough.
