We are given the function $f(x) = (-12 - 3x)(-2 + x)$, defined over $\mathbb{R}$. Let's break down this function step-by-step and analyze it to understand its properties, which are partially shown in the graph.

Step 1: Expand the function

We start by expanding $f(x) = (-12 - 3x)(-2 + x)$:

$f(x) = (-12)(-2) + (-12)(x) + (-3x)(-2) + (-3x)(x)$ $f(x) = 24 - 12x + 6x - 3x^2$ $f(x) = -3x^2 - 6x + 24$

Thus, the function is a quadratic function: $f(x) = -3x^2 - 6x + 24$

Step 2: Identify key features of the quadratic function

The general form of a quadratic function is $f(x) = ax^2 + bx + c$. In this case, we have $a = -3$, $b = -6$, and $c = 24$. This means the parabola opens downwards (because $a < 0$).

1. Vertex: The x-coordinate of the vertex is given by $x = -\frac{b}{2a}$. Plugging in the values of $a$ and $b$:

   $x = -\frac{-6}{2(-3)} = 1$

   To find the y-coordinate of the vertex, substitute $x = 1$ into the function: $f(1) = -3(1)^2 - 6(1) + 24 = -3 - 6 + 24 = 15$

   So, the vertex of the parabola is $(1, 15)$.

2. Intercepts:

   • x-intercepts: Set $f(x) = 0$ and solve for $x$: $-3x^2 - 6x + 24 = 0$ Dividing through by $-3$: $x^2 + 2x - 8 = 0$ Factoring the quadratic equation: $(x - 2)(x + 4) = 0$ So, the solutions are $x = 2$ and $x = -4$. These are the x-intercepts of the function.

   • y-intercept: To find the y-intercept, set $x = 0$: $f(0) = -3(0)^2 - 6(0) + 24 = 24$ So, the y-intercept is $(0, 24)$.

Step 3: Analyze the graph

The graph in the image shows a downward-opening parabola, which is consistent with our function. The curve intersects the x-axis at $x = -4$ and $x = 2$, as expected from our factorization. The vertex at $(1, 15)$ marks the highest point of the parabola, and the y-intercept occurs at $(0, 24)$.

Would you like more details or explanations on any of the steps?

Here are five relative questions based on the function and the graph:

1. What is the significance of the discriminant of a quadratic equation in determining the nature of its roots?
2. How would the graph change if the leading coefficient (the coefficient of $x^2$) were positive?
3. Can we determine the axis of symmetry of the parabola from the equation of the function? If so, how?
4. What are the implications of the vertex being the maximum point in the context of this function?
5. How can you verify the intercepts of the function graphically and algebraically?

Tip:

When dealing with quadratic functions, always check the sign of the leading coefficient to determine whether the parabola opens upwards or downwards.