function increasing

Question

f(x) = x^2 +6x +17

 

On what interval is the function increasing?

Answer 1

In this can, we can find the vertex by completing the square.

 

a(x + h)² + k         and      vertex = (h,k)

 

Since the leading term is x², a will be 1.   The second term is 6x.  So we need to find a binomial when squared will give us the term 6x.

 

(x + 3)² = (x + 3)(x + 3) = x² + 6x + 9

 

As for our k value, we need to use the terms 17 and 9.  Subtract these two numbers and we will get 8.  So our result is

 

(x + 3)² + 8

 

The vertex is (-3, 8).  This is because (x + 3)² is a horizontal shift to the left 3 units from the origin.

 

The vertex is the point where the graph changes from decreasing to increasing and vice versa.  Since x² is a positive parabola, the graph first decreases then increases.  -3 is the location where the change happens.

 

Therefore, the function is increasing at interval (-3, ∞).