Question

Over which interval is the graph of f(x) = x2 + 5x + 6 increasing?

Answer 1

(–5, ∞)

Explanation:

This is a vertical parabola open upward

The vertex represent a minimum

The vertex is the point (-5,-6.5)

The domain is all real numbers

The range is the interval [-6.5,∞)

so

At the left of the x-coordinate of the vertex the function is decreasing and at the right of the x-coordinate of the vertex the function is increasing

therefore

The function is increasing in the interval (-5,∞) and the function is decreasing in the interval (-∞,-5)

Answer 2

Explanation:

Given is a function

`f(x) = x^2 + 5x + 6`

Since leading term is positive,

the parabola is open up.

Let us write this in vertex form after completing the square

`f(x) = x^2 +2((5)/(2))x+(25)/(4)  -(25)/(4)+6\\=(x+((5)/(2))^2-(1)/(4)`

Hence we get vertex= (-5/2,-1/4)

Since this is a parabola open up we have the minimum point is only one at x=-5/2

So we have the function decreasing for x<-2.5 and increasing for x>-2.5

Decreases in
`(-\infty, -2.5)`

Increases in
`(-2.5,\infty)`