Question

What is the range of f(x)=-4(x+2)^2-6

Answer 1

(-∞, -6]

Explanation:

The given function is f(x) = -4(x+2)^2 - 6.

To determine the range of the function, we need to find the set of all possible output values (y-values) that the function can take.

Let's analyze the function:

• The term (x+2)^2 represents a parabolic function with its vertex at (-2, 0). The coefficient -4 indicates that the parabola opens downward.
• The term -6 shifts the entire graph downward by 6 units.

Since the parabola opens downward, the vertex represents the maximum point of the function. Thus, the range of the function is all real numbers less than or equal to the y-coordinate of the vertex.

The y-coordinate of the vertex is f(-2) = -4(-2+2)^2 - 6 = -6.

Therefore, the range of f(x) = -4(x+2)^2 - 6 is (-∞, -6]. In interval notation, this means that the range includes all real numbers less than or equal to -6.

Answer 2

y ≤ -6 or [-6, -∞)

Explanation:

f(x) = -4(x + 2)² - 6

(x + 2)² has a minimum value of 0 and is non-negative.

-4(x + 2)² has a maximum value of 0, and cannot have a positive value.

-4(x + 2)² - 6 is negative with a maximum value of -6.

Answer: y ≤ -6 or [-6, -∞)