Consider the graph of the function f(x) = 2(x + 3)2 + 2. Over which interval is the graph decreasing? (–∞, –3) (–∞, 2) (–3, ∞) (2, ∞)

The interval over which the graph is decreasing is;

(-∞, -3)

Explanation:

The given function is f(x) = 2·(x + 3)² + 2

By expanding the function, we have;

2·x² + 12·x + 20

From the characteristics of a quadratic equation, we have;

The shape of a quadratic equation = A parabola

The coefficient of x² = +2 (positive), therefore the parabola opens up

The parabola has a minimum point

Points to the left of the minimum point are decreasing

The minimum point is obtained as the x-coordinate value when f'(x) = 0

∴ f'(x) = d(2·x² + 12·x + 20)/dx = 4·x + 12

At the minimum point,  f'(x) = 4·x + 12 = 0

∴ x = -12/4 = -3

Therefore;

The graph is decreasing over the interval from -infinity (-∞) to -3 which is (-∞, -3)

Please find attached the graph of the function created with Microsoft Excel

The graph is decreasing over the interval (-∞, -3).

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