SOLUTION: For the function y = 3x^2 + 12x + 9 find: A. Whether the curve is concave up or down B. The y-intercept C. The x-intercept D. The Vertex E. The Maximum or Minimum

Question 154223: For the function y = 3x^2 + 12x + 9 find:
A. Whether the curve is concave up or down
B. The y-intercept
C. The x-intercept
D. The Vertex
E. The Maximum or Minimum value.

Answer by jim_thompson5910(35256) (Show Source): 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.

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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)

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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)

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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

.

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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

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Here's a graph to verify our answers:

Graph of
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