The given function is:

$f(x) = x^2 - 3x - 4$

This is a quadratic function, and it can be analyzed in several ways.

Key Properties:

1. Vertex: The vertex of a quadratic function $f(x) = ax^2 + bx + c$ is given by the formula: $x_{\text{vertex}} = -\frac{b}{2a}$ For $f(x) = x^2 - 3x - 4$, $a = 1$, $b = -3$, and $c = -4$, so the $x$-coordinate of the vertex is: $x_{\text{vertex}} = -\frac{-3}{2(1)} = \frac{3}{2}$ To find the $y$-coordinate of the vertex, substitute $x = \frac{3}{2}$ back into the function: $f\left(\frac{3}{2}\right) = \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) - 4 = \frac{9}{4} - \frac{9}{2} - 4 = -\frac{25}{4}$ Thus, the vertex is $\left(\frac{3}{2}, -\frac{25}{4}\right)$.

2. Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex. It is: $x = \frac{3}{2}$

3. Y-intercept: The y-intercept occurs when $x = 0$. Substitute