the graph of the function f(x)=x² + 8x + 12 is shown. which statements describe the graph? check all that…

Explanation:

Step1: Analyze the vertex

For a quadratic function $y = ax^{2}+bx + c$ ($a\neq0$), the $x$-coordinate of the vertex is $x=-\frac{b}{2a}$. For $f(x)=x^{2}+8x + 12$ where $a = 1$, $b = 8$, $x=-\frac{8}{2\times1}=-4$. Substitute $x=-4$ into $f(x)$: $f(-4)=(-4)^{2}+8\times(-4)+12=16 - 32+12=-4$. Since $a = 1>0$, the parabola opens up and the vertex is the minimum value, so the first - statement is false.

Step2: Find the axis of symmetry

The axis of symmetry of a quadratic function $y = ax^{2}+bx + c$ is $x=-\frac{b}{2a}$. Here $a = 1$, $b = 8$, so $x=-\frac{8}{2\times1}=-4$. The second - statement is true.

Step3: Determine the domain

The domain of a quadratic function $y = ax^{2}+bx + c$ is all real numbers because we can substitute any real number for $x$. So the domain of $f(x)=x^{2}+8x + 12$ is all real numbers. The third - statement is true.

Step4: Determine the range

Since the parabola opens up ($a = 1>0$) and the vertex is at $(-4,-4)$, the range is $y\geq - 4$, not all real numbers. The fourth - statement is false.

Step5: Analyze the increasing interval

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 $a = 1>0$ and $-\frac{b}{2a}=-4$, so it is decreasing on $(-\infty,-4)$ and increasing on $(-4,\infty)$. The fifth - statement is false.

Step6: Find the x - intercepts

Set $y = f(x)=x^{2}+8x + 12 = 0$. Factor the quadratic equation: $(x + 6)(x+2)=0$. Then $x=-6$ or $x=-2$. So the $x$-intercepts are at $(-6,0)$ and $(-2,0)$. The sixth - statement is true.

Answer:

The axis of symmetry is $x=-4$. The domain is all real numbers. The $x$-intercepts are at $(-6,0)$ and $(-2,0)$.