Question

Complete the square to determine the maximum or minimum value of the function defined by the expression.

Answer 1

Answer:

maximum of 15

Step-by-step explanation:

Complete the square on

   y = -x² - 6x + 6

First, factor out -1 from the first 2 terms so the x² term is positive

  y = -(x² + 6x) + 6

A quadratic equation is in the form of y = ax² + bx + c.  To complete the square, take half of the b term (here, the b term is 6), then square it...

6/2 = 3

3² = 9

Now add and subtract that from the equation...

y = -(x² - 6x + 9 - 9) + 6

Now pull out the -9 from the parenthesis, be careful though, there is a -1 multiplier in front of the parenthesis, so it come out as a positive 9

y = -(x² - 6x + 9) + 9 + 6

x² - 6x + 9 is a perfect square (we did this by completing the square), so it factors to (x - 3)², 9 + 6 = 15, so our equation becomes...

y = -(x - 3)² + 15

This is now in vertex form, which is either the minimum or maximum.

Vertex form is

y = a(x - h)² + k, where (h, k) is the vertex.  If a > 0, then the vertex is a minimum, if a < 0, then the vertex is a maximum.

We have a vertex of (3, 15) which is a maximum since a < 0.  The maximum value is the y coordinate, which is 15

The x coordinate is positive 3 because we have (x - h)² and h is 3 in this case

Answer 2

Answer:

Option B - Maximum value at 15

Step-by-step explanation:

Given : Expression $y=-x^2-6x+6$

To find : Complete the square to determine the maximum or minimum value of the function defined by the expression?

Solution :

Step 1 - Write quadratic function in standard or vertex form.

The vertex form $y=a(x-h)^2+k$

Applying completing the square into $y=-x^2-6x+6$

$y=-x^2-6x+6$

$y=-x^2-6x+6+3^2-3^2$

$y=-(x^2+3)^3+9+6$

$y=-(x^2+3)^3+15$

Step 2 - Determine the direction of the graph.

If the parabola opens upward, we will be finding its minimum value.

If the parabola opens downward, we will find its maximum value.

It depends on the value of a

a=-1 i.e. negative so the parabola opens downward we will find its maximum value.

Step 3 - The maximum value attain at point k

$y=-(x^2+3)^3+15$

(h,k) are the vertex of the parabola on comparing we get,

k=15

So, The Expression has maximum value at 15.

Therefore, Option B is correct.