analyzing a quadratic function\n\nthe graph of the function $f(x) = x^2 + 8x + 12$ is shown. which…

analyzing a quadratic function\n\nthe graph of the function $f(x) = x^2 + 8x + 12$ is shown. which statements describe the graph? check all that apply.\n\n- the vertex is the maximum value.\n- the axis of symmetry is $x = -4$.\n- the domain is all real numbers.\n- the range is all real numbers.\n- the function is increasing over $(-\\infty, -4)$.\n- the $x$-intercepts are at $(-6, 0)$ and $(-2, 0)$.

analyzing a quadratic function\n\nthe graph of the function $f(x) = x^2 + 8x + 12$ is shown. which statements describe the graph? check all that apply.\n\n- the vertex is the maximum value.\n- the axis of symmetry is $x = -4$.\n- the domain is all real numbers.\n- the range is all real numbers.\n- the function is increasing over $(-\\infty, -4)$.\n- the $x$-intercepts are at $(-6, 0)$ and $(-2, 0)$.

Answer

Explanation:

Step1: Determine vertex type

The parabola opens upward ($a=1 > 0$), so the vertex is a minimum.

Step2: Find axis of symmetry

Using $x = -\frac{b}{2a}$ for $f(x) = x^2 + 8x + 12$: $$x = -\frac{8}{2(1)} = -4$$

Step3: Identify the domain

Quadratic functions are defined for all real values of $x$. $$\text{Domain: } (-\infty, \infty)$$

Step4: Identify the range

The minimum value is $f(-4) = (-4)^2 + 8(-4) + 12 = -4$. $$\text{Range: } [-4, \infty)$$

Step5: Determine increasing interval

The function increases to the right of the vertex ($x > -4$). $$\text{Increasing over: } (-4, \infty)$$

Step6: Find x-intercepts

Set $f(x) = 0$: $x^2 + 8x + 12 = (x+6)(x+2) = 0$. $$x = -6, x = -2$$

Answer:

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