Question

for the quadratic function f(x)= -x² - 2x, answer parts (a) through (f). the domain of f is (-∞,∞). (type your answer in interval notation.) the range of f is (-∞,1. (type your answer in interval notation.) (e) determine where the quadratic function is increasing and where it is decreasing. the function is increasing on the interval . (type your answer in interval notation.)

Explanation:

Step1: Rewrite the function in vertex - form

For a quadratic function $y = ax^{2}+bx + c$, the vertex - form is $y=a(x - h)^{2}+k$, where the vertex is $(h,k)$. Given $f(x)=-x^{2}-2x$, we complete the square:
\[

$$\begin{align*} f(x)&=-(x^{2}+2x)\\ &=-(x^{2}+2x + 1-1)\\ &=-(x + 1)^{2}+1 \end{align*}$$

\]
Here, $a=-1$, $h=-1$, $k = 1$.

Step2: Analyze the direction of the parabola

Since $a=-1<0$, the parabola opens downwards.

Step3: Determine the increasing interval

For a parabola $y = a(x - h)^{2}+k$ that opens downwards ($a<0$), the function is increasing on the interval $(-\infty,h)$.
Since $h=-1$, the function is increasing on the interval $(-\infty,-1)$.

Answer:

$(-\infty,-1)$