7. find the vertrex about \( y=x^{2}-3 x-4 \)

To find the vertex of the parabola given by the equation \( y = x^2 - 3x - 4 \), we can use the formula for the x-coordinate of the vertex, which is \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -3 \). Plugging in these values, we find \( x = -\frac{-3}{2(1)} = \frac{3}{2} \). Now, substitute \( x = \frac{3}{2} \) back into the equation to find the y-coordinate: \( y = \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) - 4 = \frac{9}{4} - \frac{9}{2} - 4 = \frac{9}{4} - \frac{18}{4} - \frac{16}{4} = -\frac{25}{4} \). Thus, the vertex is at \( \left(\frac{3}{2}, -\frac{25}{4}\right) \). For a little color, did you know that parabolas describe the trajectory of projectiles? Much like how a basketball arcs toward the hoop, the vertex represents the peak point or minimum point depending on whether the parabola opens upward or downward. If you're solving these problems and want to avoid common pitfalls, remember to double-check your signs for \( a \) and \( b \) when using the vertex formula. It’s easy to mix them up—like trying to find the best pizza place but getting lost on the way!