Question

(1 point)
Responses
Using graphing technology to find the range of the function f(x) = -x^2 - 4x -4
(−∞,0)
(−∞,0)

(0,0)
(0,0)

(−∞,0]
(−∞,0]

All real numbers

Answer 1

To find the range of the function \( f(x) = -x^2 - 4x - 4 \), we first note that this is a quadratic function, which opens downwards since the coefficient of \( x^2 \) is negative.

To determine the maximum value of this function, we can complete the square or use the vertex formula for a quadratic function.

The vertex form of a quadratic function is given by:

\[
f(x) = a(x-h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola. The \(x\)-coordinate of the vertex can be found using:

\[
h = -\frac{b}{2a}
\]

For the quadratic \(f(x) = -x^2 - 4x - 4\), we have:

- \(a = -1\)
- \(b = -4\)

Now, calculate \(h\):

\[
h = -\frac{-4}{2 \cdot -1} = -\frac{4}{-2} = 2
\]

Now we can calculate the value of \(f\) at \(x = -2\):

\[
f(-2) = -(-2)^2 - 4(-2) - 4 = -4 + 8 - 4 = 0
\]

The vertex of the parabola is at the point \((-2, 0)\). Since the parabola opens downward, the highest value (maximum value) of \(f(x)\) is \(0\) which occurs at \(x = -2\).

The function goes to \(-\infty\) as \(x\) moves away from \(-2\) in either direction.

Thus, the range of the function is all values less than or equal to \(0\):

\[
\text{Range} = (-\infty, 0]
\]

So, the correct response is:

**(−∞, 0]**