You can
put this solution on YOUR website!y = 3x^2 + 12x + 9 ... Start with the given equation.
y' = 6x + 12 ... Find the first derivative
y'' = 6 ... Find the second derivative
Since y'' is
always positive, this means that the function is concave up.
-------------
b)
Y-Intercept:
Start with the given equation.
Plug in
.
Square
to get
.
Multiply
and
to get
.
Multiply
and
to get
.
Combine like terms.
So the y-intercept is (0,9)
----------------------
c)
X-Intercept(s):
Start with the given equation.
Plug in
Notice we have a quadratic equation in the form of
where
,
, and
Let's use the quadratic formula to solve for x
Start with the quadratic formula
Plug in
,
, and
Square
to get
.
Multiply
to get
Subtract
from
to get
Multiply
and
to get
.
Take the square root of
to get
.
or
Break up the expression.
or
Combine like terms.
or
Simplify.
So the answers are
or
So the x-intercepts are (-1,0) and (-3,0)
----------------------------
d)
Vertex:
In order to find the vertex, we first need to find the x-coordinate of the vertex.
To find the x-coordinate of the vertex, use this formula:
.
Start with the given formula.
From
, we can see that
,
, and
.
Plug in
and
.
Multiply 2 and
to get
.
Divide.
So the x-coordinate of the vertex is
. Note: this means that the axis of symmetry is also
.
Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.
Start with the given equation.
Plug in
.
Square
to get
.
Multiply
and
to get
.
Multiply
and
to get
.
Combine like terms.
So the y-coordinate of the vertex is
.
So the vertex is
)
.
----------------------------
e)
Maximum or Minimum value:
Since the function is concave up, this means that the function has a minimum. The max/min value correspond to the y coordinate of the vertex. So the minimum value is
-----------------------------
Here's a graph to verify our answers:
Graph of
(