the graph of the function f(x)=(x + 6)(x + 2) is shown. which statements describe the graph? check all that…

Explanation:

Step1: Expand the function

$f(x)=(x + 6)(x + 2)=x^{2}+8x + 12$. For a quadratic function $y = ax^{2}+bx + c$ ($a\neq0$), here $a = 1$, $b = 8$, $c = 12$.

Step2: Find the axis - of - symmetry

The formula for the axis of symmetry of a quadratic function $y=ax^{2}+bx + c$ is $x=-\frac{b}{2a}$. Substituting $a = 1$ and $b = 8$ gives $x=-\frac{8}{2\times1}=-4$. So the axis of symmetry is $x = - 4$, and the second statement is correct.

Step3: Determine the nature of the vertex

Since $a = 1>0$, the parabola opens upward, and the vertex is the minimum value, so the first statement is incorrect.

Step4: Find the domain

The domain of a quadratic function is all real numbers. So the domain of $y=x^{2}+8x + 12$ is all real numbers, and the third statement is correct.

Step5: Analyze the increasing and decreasing intervals

For a parabola $y = ax^{2}+bx + c$ with $a>0$, it is decreasing on $(-\infty,-\frac{b}{2a})$ and increasing on $(-\frac{b}{2a},\infty)$. Here, it is decreasing on $(-\infty,-4)$ and increasing on $(-4,\infty)$, so the fourth statement is incorrect.

Step6: Analyze where the function is negative

Set $y=(x + 6)(x + 2)=0$, the roots are $x=-6$ and $x=-2$. Since the parabola opens upward, the function is negative between the roots, i.e., over the interval $(-6,-2)$. So the fifth statement is correct.

Answer:

The axis of symmetry is $x=-4$. The domain is all real numbers. The function is negative over $(-6,-2)$.