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

To find the vertex of the parabola represented by \( y = x^2 + 3x - 4 \), you can use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = 3 \). Plugging in the numbers gives \( x = -\frac{3}{2 \cdot 1} = -\frac{3}{2} \). To find the \( y \)-coordinate, substitute \( x = -\frac{3}{2} \) back into the equation, giving \( y = \left(-\frac{3}{2}\right)^2 + 3\left(-\frac{3}{2}\right) - 4 = -\frac{25}{4} \). So, the vertex is at \( \left(-\frac{3}{2}, -\frac{25}{4}\right) \). Understanding the vertex of a parabola is not just a mathematical exercise; it has real-world applications in fields like physics and engineering. For instance, the vertex can represent the maximum or minimum point of a projectile's trajectory, determining where it reaches the peak height. This concept is crucial for optimizing designs, like sports equipment or vehicle aerodynamics, where maximizing height or distance is essential. When calculating the vertex, common mistakes often stem from miscalculating the coordinates or not substituting values correctly. It's easy to confuse the signs when working with the formula or make arithmetic errors after substituting the \( x \)-value back into the equation. Double-checking each step and ensuring proper arithmetic can save time and frustration in obtaining the correct vertex!