Question

identify the graph of $f(x)=-(x - 3)(x + 1)$.

Explanation:

Step1: Find the x - intercepts

Set $f(x)=0$, then $-(x - 3)(x + 1)=0$. Using the zero - product property, $x-3 = 0$ or $x + 1=0$, so $x = 3$ and $x=-1$ are the x - intercepts.

Step2: Determine the shape of the parabola

The function $f(x)=-(x - 3)(x + 1)=-x^{2}+2x + 3$ is a quadratic function with $a=-1<0$. So the parabola opens downwards.

Step3: Find the vertex

The x - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$ is $x=-\frac{b}{2a}$. For $y=-x^{2}+2x + 3$, $a=-1$ and $b = 2$, so $x=-\frac{2}{2\times(-1)} = 1$. Substitute $x = 1$ into $y=-x^{2}+2x + 3$, we get $y=-1^{2}+2\times1+3=4$. So the vertex is $(1,4)$.

The graph with x - intercepts at $x=-1$ and $x = 3$, opening downwards and vertex at $(1,4)$ is the correct one.

Answer:

The graph that has x - intercepts at $x=-1$ and $x = 3$, opens downwards and has a vertex at the point $(1,4)$. (Since no specific graph is labeled among the options in the description, this is a general description of the correct graph).