Question

On a coordinate plane, a parabola opens up. It goes through (negative 8, negative 2), has a vertex at (negative 5, negative 6.5), goes through (negative 2, negative 2), and has a y-intercept at (0, 6). Over which interval is the graph of f(x) = one-halfx2 + 5x + 6 increasing? (–6.5, ∞) (–5, ∞) (–∞, –5) (–∞, –6.5)

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

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

Explanation:

we have

`f(x)=(1)/(2)x^2+5x+6`

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)