Find the vertex of g(x) = 3x2 + 12x + 15

Vertex:

(

(

2

,

3

)

)

f

(

x

)

-intercept:

15

x

-intercepts: (no Real intercepts); Complex intercepts

2

±

i

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:

the line of