Answer: Vertex:
(
(
2
,
3
)
)
f
(
x
)
-intercept:
15
x
-intercepts: (no Real intercepts); Complex intercepts
2
±
i
Step-by-step explanation:
To find the vertex convert the given equation into vertex form.
f
(
x
)
=
3
x
2
−
12
x
+
15
XXX
f
(
x
)
=
3
(
x
2
−
4
x
)
+
15
XXX
f
(
x
)
=
3
(
x
2
−
4
x
+
4
)
+
15
−
3
×
4
XXX
f
(
x
)
=
3
(
x
−
2
)
2
+
3
which is the vertex form for a parabola with vertex at
(
2
,
3
)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The
f
(
x
)
-intercept is the value of
f
(
x
)
when
x
=
0
Using the original form of the equation we can see that
XXX
f
(
x
)
=
15
when
x
=
0
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The
x
-intercept(s) is/are the value of
x
when
f
(
x
)
=
0
.
Using the vertex form we derived
XXX
0
=
3
(
x
−
2
)
2
+
3
XXX
→
(
x
−
2
)
2
+
1
=
0
XXX
→
(
x
−
2
)
2
=
−
1
Since the square of any Real number is greater than or equal to zero
we can see that this equation has no Real solutions.
However if Complex solutions are permitted
XXX
x
−
2
=
±
√
−
1
=
±
i
XXX
→
x
=
2
±
i
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Step-by-step explanation: