function $f$ is given by $f(x)=(x - 2)(x + 4)$. without using graphing technology, answer the following questions.\n1. what are the $x$-intercepts of the graph representing $f$?\n2. what are the $x$- and $y$-coordinates of the vertex of the graph?\n3. what is the $y$-intercept?\n4. sketch a graph that represents $f$.

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To find the vertex of the parabola, we use the fact that the x-coordinate of the vertex is the midpoint between the x-intercepts. Since our x-intercepts are at x = -4 and x = 2, the x-coordinate of the vertex is (-4 + 2) divided by 2, which equals -1. To find the y-coordinate, we substitute x = -1 into our function: f(-1) = (-1 - 2)(-1 + 4) = (-3)(3) = -9. Therefore, the vertex is at the point (-1, -9). To find the y-intercept, we need to determine where the graph crosses the y-axis. This occurs when x equals zero. We substitute x = 0 into our function: f(0) = (0 - 2)(0 + 4) = (-2)(4) = -8. Therefore, the y-intercept is at the point (0, -8). Here is the complete graph of f(x) = (x - 2)(x + 4). The parabola opens upward because the leading coefficient is positive. It crosses the x-axis at (-4, 0) and (2, 0), reaches its minimum at the vertex (-1, -9), and crosses the y-axis at (0, -8). The axis of symmetry is the vertical line x = -1, which passes through the vertex. This quadratic function represents a U-shaped curve with all the key features we calculated. In summary, we have completely analyzed the quadratic function f(x) = (x - 2)(x + 4) without using graphing technology. We found the x-intercepts by setting the function equal to zero, giving us the points (-4, 0) and (2, 0). The vertex was located at (-1, -9) by finding the midpoint of the x-intercepts and evaluating the function at that point. The y-intercept was found by substituting x = 0, resulting in the point (0, -8). Finally, we sketched an upward-opening parabola that passes through all these key points, with the axis of symmetry at x = -1. This systematic approach allows us to fully understand the behavior of any quadratic function in factored form.