Question

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

Explanation:

Step1: Find the x - intercepts

Set $f(x)=0$, then $(x - 3)(x + 1)=0$. By the zero - product property, $x-3 = 0$ gives $x = 3$ and $x+1=0$ gives $x=-1$. So the x - intercepts are $x = 3$ and $x=-1$.

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 positive leading coefficient ($a = 1$ in $y=ax^{2}+bx + c$ form), so the parabola opens upward.

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 $x=-\frac{-2}{2\times1}=1$. Substitute $x = 1$ into the function: $y=(1 - 3)(1 + 1)=-4$. The vertex is $(1,-4)$.

Answer:

The graph with x - intercepts at $x=-1$ and $x = 3$ and vertex at $(1,-4)$ and opening upward. (Since no specific labels for the graphs are given in text, the description based on key points is the answer. If there were labels like Graph A, Graph B etc., we would identify the corresponding one based on these characteristics).