which statements about the graph of the function (f(x)=-x^{2}-4x + 2) are true? choose three correct…

Explanation:

Step1: Find the vertex of the parabola

The function $f(x)=-x^{2}-4x + 2$ is in the form $y = ax^{2}+bx + c$ where $a=-1$, $b=-4$, $c = 2$. The $x$-coordinate of the vertex is $x=-\frac{b}{2a}=-\frac{-4}{2\times(-1)}=-2$. Substitute $x = - 2$ into the function: $f(-2)=-(-2)^{2}-4\times(-2)+2=-4 + 8+2=6$. So the vertex is $(-2,6)$. Since $a=-1<0$, the parabola opens down - ward.

Step2: Analyze the increasing - decreasing intervals

The function is increasing on the interval $(-\infty,-2)$ and decreasing on the interval $(-2,\infty)$.

Step3: Find the $y$-intercept

Set $x = 0$ in the function: $f(0)=-0^{2}-4\times0 + 2=2>0$. So the $y$-intercept is positive.

Step4: Determine the domain and range

The domain of a quadratic function $y = ax^{2}+bx + c$ is all real numbers, i.e., ${x|x\in R}$. Since the vertex is $(-2,6)$ and the parabola opens downward, the range is ${y|y\leq6}$.

Answer:

The function is increasing over the interval $(-\infty,-2)$. The function has a positive $y$-intercept. The range is ${y|y\leq6}$.